Find the probability that two kings are selected. A coin is tossed 4 times. Find the probability of getting 4 heads. A die is rolled and a card is selected at random from a deck of 52 cards.
Find the probability of getting an odd number on the die and a club on the card. The probability of selecting 3 adults who are left-handed is 0. Find the probability of getting two kings. The second multiplication rule follows. Since the second toaster is selected from the remaining 23 and there are two defective toasters left, the probability that 2 it is defective is Find the probability that both are queens. Three balls are selected without replacement.
Find the probability of selecting 2 yellow balls and a white ball. It means to multiply. In a study, there are 8 guinea pigs; 5 are black and 3 are white.
In a classroom there are 8 freshmen and 6 sophomores. Three cards are drawn from a deck of 52 cards without replacement. Find the probability of getting 3 diamonds. A box contains 12 calculators of which 5 are defective. Three are dead. Recall that P B A means the probability of event B occurring given that event A has already occurred.
Another situation where conditional probability can be used is when additional information about an event is known. When conditions are imposed or known on events, there is a possibility that the probability of the certain event occurring may change. For example, suppose you want to determine the probability that a house will be destroyed by a hurricane. If you used all houses in the United States as the sample space, the probability would be very small. However, if you used only the houses in the states that border the Atlantic Ocean as the sample space, the probability would be much higher.
Consider the following examples. Hence the probability of getting a 4 is 13 since there is one chance in three of getting a 4 if it is known that the result was an even number. Find the probability of getting a sum of 3 if it is known that the sum of the spots on the dice was less than six. They are 1, 2 and 2, 1 , and there are 10 ways to get a sum less than six.
The two previous examples of conditional probability were solved using classical probability and reduced sample spaces; however, they can be solved by using the following formula for conditional probability. The two previous examples will now be solved using the formula for conditional probability.
Notice that there is only one way to get a 4 and an even number—the outcome 4. Find the probability of getting a sum of 3 if it is known that the sum of the spots on the dice was less than 6. There are 5 ways to get a sum of 8; hence, P sum of 8 sum is 8 even is 18 or Find the probability that a house has a deck given that it has a garage. Find the probability of getting two tails if it is known that one of the coins is a tail. Find the probability that it is an ace given that it is a black card.
The probability that a family visits the Rainbow Gardens Amusement Park is 0. Three dice are rolled. Find the probability of getting three twos if it is known that the sum of the spots of the three dice was six. There are two ways to get a sum of 8 when one die is a 6. There are two possible ways one die is a 6 and the sum of the dice is 8. Hence, the probability is There is one way to get two tails; hence, the probability of getting two tails given that one of the coins is a tail is If the events are independent, multiplication rule I is used.
If the events are dependent, multiplication rule II is used. Which of the following events are dependent? Tossing a coin and rolling a die b.
Rolling a die and then rolling a second die c. Sitting in the sun all day and getting sunburned d. Drawing a card from a deck and rolling a die 2. What is the probability of getting three 4s? What is the probability of selecting 4 spades from a deck of 52 cards if each card is replaced before the next one is selected? What is the probability of getting 5 twos? A coin is tossed four times; what is the probability of getting 4 heads? In a sample of 10 telephones, 4 are defective. If 3 are selected at random and tested, what is the probability that all will be non- defective?
A bag contains 4 blue marbles and 5 red marbles. If 2 marbles are selected at random without replacement, what is the probability that both will be blue? The numbers 1 to 12 are placed in a hat, and a number is selected. What is the probability that the number is 4 given that it is known to be an even number?
Three coins are tossed; what is the probability of getting 3 heads if it is known that at least two heads were obtained? If a person is selected at random, what is the probability that the person takes Vitamin E given that the person takes Vitamin C?
Two dice are tossed; what is the probability that the numbers are the same on both dice if it is known that the sum of the spots is 6? Most people would bet on a tail. A coin is an inanimate object. It does not have a memory. So does that make the law of averages wrong? If you played the game, you would have only one chance in of winning, but if people played the game, there is a pretty good chance that somebody would win the prize. If people played the game, there would be a good chance that two people might win.
So what does this mean? It means that the probability of winning big in a lottery or on a slot machine is very small, but since there are many, many people playing, somebody will probably win; however, your chances of winning big are very small.
A similar situation occurs when couples have children. Suppose a husband and wife have four boys and would like to have a girl. It is incorrect to reason 1 that the chance of having a family of 5 boys is 32 , so it is more likely that the next child will be a girl. However, after each child is born, the probability that the next child is a girl or a boy for that matter is about The law of averages is not appropriate here. Emery Landon and their 13 boys! Another area where people incorrectly apply the law of averages is in attempting to apply a betting system to gambling games.
One such system is doubling your bet when you lose. Consider a game where a coin is tossed. If it lands heads, you win what you bet. If it lands tails, you lose. If you get tails, you lose one dollar and bet two dollars on the next toss. If you get a tail on the second toss, you bet four dollars on the third toss.
If you win, you start over with a one dollar bet, but if you lose, you bet eight dollars on the next toss. With this system, you win every time you get a head. Runs do occur and when they do, hope that they are in your favor. He lost on the twenty-ninth roll. The event took about one hour and twenty minutes. In in a casino in Puerto Rico at a roulette game, the number 10 occurred six times in succession. There are 38 numbers on a roulette wheel. At a casino in New York in the color red occurred in a roulette game 32 times in a row, and at a casino in Monte Carlo an even number occurred in a roulette game 28 times in a row.
So what can be concluded? First, rare events events with a small probability of occurring can and do occur. Second, the more people who play a game, the more likely someone will win. Finally, the law of averages applies when there is a large number of independent outcomes in which the probability of each outcome occurring does not change. They are odds and expectation. Odds are used most often in gambling games at casinos and racetracks, and in sports betting and lotteries.
Mathematical expectation can be thought of more or less as an average over the long run. In other words, if you would perform a probability experiment many times, the expectation would be an average of the outcomes. For example, at a race, the odds that a horse wins the race may be 4 to 1. Odds are computed from probabilities. For example, suppose you roll a die and if you roll a three, you win. If you roll any other number, you lose.
In this case, there are six outcomes, and you have one chance outcome of winning, so the probability that you win is That means on average you win once in every six rolls. Of course, there is no guarantee that you will win on the sixth roll.
In gambling games, the odds are expressed backwards. For example, if there is one chance in six that you will win, the odds are 1 to 5, but in general, the odds would be given as 5 to 1. Odds can be expressed as a fraction, 15, or as a ratio, 1 : 5. If the odds of winning the game are 1 : 5, then the odds of losing are 5 : 1.
If the odds in favor of an event occurring are A : B, then the odds against the event occurring are B : A. For example, if the odds are 1 : 15 that an event will occur, then the odds against the event occurring are 15 : 1.
Odds can be other numbers, such as 2 : 5, 7 : 4, etc. There is only one way to get a sum of 12, and that is 6, 6. There are 1 36 outcomes in the sample space. When three coins are tossed, there are three ways to get two tails and a head. There are six ways to get a sum of 7 and 36 outcomes in the sample 6 space. Previously it was shown that given the probability of an event, the odds in favor of the event occurring or the odds against the event occurring can be found.
The opposite is also true. Find the probability that an event E will occur if the odds are in favor of E. Find the probability that an event E will not occur if the odds against the event E are 4 : 1.
Find the probability that an event E will occur if the odds in favor of the event are 2 : 3. Expectation or expected value is a long run average. The expected value is also called the mean, and it is used in games of chance, insurance, and in other areas such as decision theory. The outcomes must be numerical in nature. The expected value of the outcome of a probability experiment can be found by multiplying each outcome by its corresponding probability and adding the results.
Now what does this mean? When a die is rolled, it is not possible to get 3. In other words, 3. That would make the game fair. The probability of getting no heads is Winning amounts are positive and losses are negative.
Find the expected value of the game if a person buys one ticket. The probability of winning is one chance in since tickets are sold. There is an alternative method that can be used to solve problems when tickets are sold or when people pay to play a game. In this case, multiply the prize value by the probability of winning and subtract the cost of the ticket or the cost of playing the game.
When the expected value of a game is negative, it is in favor of the house i. When the expected value of a game is positive, it is in favor of the player. The last situation rarely ever happens unless the con man is not knowledgeable of probability theory. Find the expected value if a person purchases 2 tickets.
Consider the game called Chuck-a-luck. Then three dice are tossed usually in a cage. Con men like to say that the probability of any number coming up is 16 on each die; therefore, each number has a probability of 36 or 12 of occurring, and if it occurs more than once, the player wins more money.
Hence, the game is in favor of the player. This is not true. The next example shows how to compute the expected value for the game of Chuck-a-luck. Find the expected value if a person purchased one ticket. How much should the person pay if the game is to be fair? There is one way to roll a sum of two and one way to roll a sum of 12; there are 36 outcomes in the sample space. Odds are computed from probabilities; how- ever, probabilities can be computed from odds if the true odds are known.
Mathematical expectation can be thought of more or less as a long run average. If the game is played many times, the average of the outcomes or the payouts can be computed using mathematical expectation. Three coins are tossed. What are the odds in favor of getting 3 heads?
When two dice are rolled, what are the odds against getting doubles? When a card is selected from a deck of 52 cards, what are the odds in favor of getting a face card? When a die is rolled, what are the odds in favor of getting a 5 or a 6? On a roulette wheel, there are 38 numbers: 18 numbers are red, 18 numbers are black, and 2 are green.
What are the odds in favor of getting a red number when the ball is rolled? If the odds in favor of an event occurring are , what are the odds against the event occurring? If the odds against an event occurring are , what are the odds in favor of the event occurring? The probability of an event occurring is What are the odds in favor of the event occurring?
What are the odds against the event occurring? What are the odds for a fair game? When a game is fair the expected value would be a. When three coins are tossed, what is the expected value of the number of heads? A special die is made with 1 one, 2 twos, 3 threes. What is the expected number of spots for one roll? What is the expected value if a person purchases one ticket?
A person selects one bill at random and wins that bill. How much should the person pay to play the game if it is to be fair? Mendel lived in a monastery all of his adult life and based his research on the observation of plants.
He published his results in an obscure journal and the results remained unknown until the beginning of the 20th century. At that time, his research was used by a mathematician, G. Hardy, to study human genetics. Genetics is somewhat more complicated than what is presented here.
How- ever, what is important here is to explain how probability is used in genetics. There were two colors, yellow and green. Mendel theorized that each egg cell and each pollen cell contained two color genes that split on fertilization. There were three possibilities: pure yellow seeds, pure green seeds, and hybrid-yellow seeds. The pure yellow seeds contain two yellow genes. The pure green seeds contained two green genes. The hybrid-yellow seeds contain one yellow gene and one green gene.
This seed was yellow since the yellow gene is dominant over the green gene. The green gene is said to be recessive. Next consider the possibilities. If there are two pure yellow plants, then the results of fertilization will be YY as shown.
What happens with a pure yellow and a pure green plant? For example, for the gender of children, the female egg contains two X chromosomes, and the male contains X and Y chromosomes. This is especially true when the number of outcomes is large.
This computation will be shown later in this chapter. In order to do the computation, we use the fundamental counting rule, the permutation rules, and the combination rule. Finally, a donor can be male or female. A license plate consists of 2 letters and 3 digits. How many can be made if repetitions are not permitted? Some examples of factorial notation are 6! For example, 6! The card is a king 2. The card is a diamond 3. The card is a king and a diamond.
That is, the card is the king of diamonds. In the second example, we say the two events are not mutually exclusive. Two events then are mutually exclusive if they cannot occur at the same time. In other words, the events have no common outcomes. Selecting a card at random from a deck and getting an ace or a club Rolling a die and getting an odd number or a number less than 4 Rolling two dice and getting a sum of 7 or 11 Selecting a student at random who is full-time or part-time Selecting a student who is a female or a junior SOLUTION: a.
The ace of clubs is an outcome of both events. One and three are common outcomes. Yes Yes No. A female student who is a junior is a common outcome. Find the probability that the card is an ace or a king. In a box there are 3 red pens, 5 blue pens, and 2 black pens. A blue or a red pen. A red or a black pen. A small automobile dealer has 4 Buicks, 7 Fords, 3 Chryslers, and 6 Chevrolets.
A Buick or a Chevrolet. A Chrysler or a Chevrolet. In a model railroader club, 23 members model HO scale, 15 members model N scale, 10 members model G scale, and 5 members model O scale. N or G scale. HO or O scale. A package of candy contains 8 red pieces, 6 white pieces, 2 blue pieces, and 4 green pieces.
White or green. Blue or red. On a bookshelf in a classroom there are 6 mathematics books, 5 reading books, 4 science books, and 10 history books. A history book or a mathematics book.
A reading book or a science book. In this case, addition rule II can be used. Find the probability that it is a 6 or a diamond. Since there is one card that is both a 6 and a diamond i. Find the probability of getting an even number or a number less than 4. There is only one way for this event to 1. Six of the Democrats are females and 5 of the Republicans are females. If the probability that a student owns both a computer and an automobile is 0. If the events are not mutually exclusive, the probability of the outcomes that the two events have in common must be subtracted from the sum of the probabilities of the two events.
A die is rolled. Find the probability that the result is an even number or a number less than 3. Find the probability that a number on one die is a six or the sum of the spots is eight. A coin is tossed and a die is rolled. Find the probability that the coin falls heads up or that there is a 4 on the die. In a psychology class, there are 15 sophomores and 18 juniors. Six of the sophomores are males and 10 of the juniors are males. A junior or a male. A sophomore or a female.
A junior. In these cases, the addition rules are used. When the events are mutually exclusive, addition rule I is used, and when the events are not mutually exclusive, addition rule II is used. If the events are mutually exclusive, they have no outcomes in common.
When the two events are not mutually exclusive, they have some common outcomes. Which of the two events are not mutually exclusive? Rolling a die and getting a 6 or a 3 Drawing a card from a deck and getting a club or an ace Tossing a coin and getting a head or a tail Tossing a coin and getting a head and rolling a die and getting an odd number 2.
Which of the two events are mutually exclusive? Drawing a card from a deck and getting a king or a club Rolling a die and getting an even number or a 6 Tossing two coins and getting two heads or two tails Rolling two dice and getting doubles or getting a sum of eight 3.
In a box there are 6 white marbles, 3 blue marbles, and 1 red marble. If a marble is selected at random, what is the probability that it is red or blue? When a single die is rolled, what is the probability of getting a prime number 2, 3, or 5?
A card is selected from a deck of 52 cards. Find the probability that it is a red queen or a black ace. At a high school with students, 62 play football, 33 play baseball, and 14 play both sports. A card is selected from a deck. Find the probability that it is a face card or a diamond. A single card is selected from a deck. Find the probability that it is a queen or a black card.
What is the probability of getting doubles or a sum of 10? The probability that a family visits Safari Zoo is 0. Pleasant Tourist Railroad is 0. The probability that a family does both is 0. Find the probability that the family visits the zoo or the railroad. If a card is drawn from a deck, what is the probability that it is a king, queen, or an ace?
Do you think you are more likely to win a large lottery and become a millionaire or are you more likely to be struck by lightning? Consider each probability. In a recent article, researchers estimated that the chance of winning a million or more dollars in a lottery is about one in 2 million. In a recent Pennsylvania State Lottery, the chances of winning a million dollars were 1 in 9. Now the chances of being struck by lightning are about 1 in , Thus, a person is at least three times more likely to be struck by lightning than win a million dollars!
But wait a minute! Statisticians are critical of these types of comparisons, since winning the lottery is a random occurrence. But being struck by lightning depends on several factors. For example, if a person lives in a region where there are a lot of thunderstorms, his or her chances of being struck increase. So be wary of such comparisons.
As the old saying goes, you cannot compare apples and oranges. This chapter will show you how to use the multiplication rules to solve many problems in probability. In addition, the concept of independent and dependent events will be introduced. Another example would be selecting a card from a deck, replacing it, and then selecting a second card from a deck.
For example, suppose a card is selected from a deck and not replaced, and a second card is selected. Another example would be parking in a no parking zone and getting a parking ticket. Again, if you are legally parked, the chances of getting a parking ticket are pretty close to zero as long as the meter does not run out. However, if you are illegally parked, your chances of getting a parking ticket dramatically increase.
Tossing a coin and selecting a card from a deck 2. Driving on ice and having an accident 3. Drawing a ball from an urn, not replacing it, and then drawing a second ball 4. Having a high I. In most cases, driving on ice will increase the probability of having an accident.
From classical probability theory, it can be determined that the probability of getting two heads is 14, since there is only one way to get two heads and there are four outcomes in the sample space. However, there is another way to determine the probability of getting two heads.
In other words, when two independent events occur in sequence, the probability that both events will occur can be found by multiplying the probabilities of each individual event. The word and is the key word and means that both events occur in sequence and to multiply.
Find the probability of getting a tail on the coin and a 5 on the die. A ball is selected at random and its color is noted. Then it is replaced and another ball is selected and its color is noted. Find the probability of each of these: a. Selecting 2 blue balls b. Selecting a blue ball and then a red ball c.
Find the probability of getting three 6s. Even though the subjects are not replaced, the probability changes only slightly, so the change can be ignored. Consider the next example. A card is drawn from a deck, then replaced, and a second card is drawn. Find the probability that two kings are selected. A coin is tossed 4 times. Find the probability of getting 4 heads. A die is rolled and a card is selected at random from a deck of 52 cards.
Find the probability of getting an odd number on the die and a club on the card. The probability of selecting 3 adults who are left-handed is 0. Find the probability of getting two kings. The second multiplication rule follows. Since the second toaster is selected from the remaining 23 and there are two defective toasters left, the probability that 2 it is defective is Find the probability that both are queens.
Three balls are selected without replacement. Find the probability of selecting 2 yellow balls and a white ball. It means to multiply. In a study, there are 8 guinea pigs; 5 are black and 3 are white. In a classroom there are 8 freshmen and 6 sophomores. Three cards are drawn from a deck of 52 cards without replacement.
Find the probability of getting 3 diamonds. A box contains 12 calculators of which 5 are defective. Three are dead. Recall that P B A means the probability of event B occurring given that event A has already occurred. Another situation where conditional probability can be used is when additional information about an event is known.
When conditions are imposed or known on events, there is a possibility that the probability of the certain event occurring may change. For example, suppose you want to determine the probability that a house will be destroyed by a hurricane. If you used all houses in the United States as the sample space, the probability would be very small.
However, if you used only the houses in the states that border the Atlantic Ocean as the sample space, the probability would be much higher. Consider the following examples. Hence the probability of getting a 4 is 13 since there is one chance in three of getting a 4 if it is known that the result was an even number. Find the probability of getting a sum of 3 if it is known that the sum of the spots on the dice was less than six. They are 1, 2 and 2, 1 , and there are 10 ways to get a sum less than six.
The two previous examples of conditional probability were solved using classical probability and reduced sample spaces; however, they can be solved by using the following formula for conditional probability. The two previous examples will now be solved using the formula for conditional probability. Notice that there is only one way to get a 4 and an even number—the outcome 4. Find the probability of getting a sum of 3 if it is known that the sum of the spots on the dice was less than 6.
There are 5 ways to get a sum of 8; hence, P sum of 8 sum is 8 even is 18 or Find the probability that a house has a deck given that it has a garage. Find the probability of getting two tails if it is known that one of the coins is a tail.
Find the probability that it is an ace given that it is a black card. The probability that a family visits the Rainbow Gardens Amusement Park is 0. Three dice are rolled.
Find the probability of getting three twos if it is known that the sum of the spots of the three dice was six. There are two ways to get a sum of 8 when one die is a 6. There are two possible ways one die is a 6 and the sum of the dice is 8. Hence, the probability is There is one way to get two tails; hence, the probability of getting two tails given that one of the coins is a tail is If the events are independent, multiplication rule I is used.
If the events are dependent, multiplication rule II is used. Which of the following events are dependent? Tossing a coin and rolling a die Rolling a die and then rolling a second die Sitting in the sun all day and getting sunburned Drawing a card from a deck and rolling a die 2. What is the probability of getting three 4s? What is the probability of selecting 4 spades from a deck of 52 cards if each card is replaced before the next one is selected?
What is the probability of getting 5 twos? A coin is tossed four times; what is the probability of getting 4 heads? In a sample of 10 telephones, 4 are defective. If 3 are selected at random and tested, what is the probability that all will be nondefective? A bag contains 4 blue marbles and 5 red marbles. If 2 marbles are selected at random without replacement, what is the probability that both will be blue?
The numbers 1 to 12 are placed in a hat, and a number is selected. What is the probability that the number is 4 given that it is known to be an even number? Three coins are tossed; what is the probability of getting 3 heads if it is known that at least two heads were obtained? If a person is selected at random, what is the probability that the person takes Vitamin E given that the person takes Vitamin C?
Two dice are tossed; what is the probability that the numbers are the same on both dice if it is known that the sum of the spots is 6? Most people would bet on a tail. A coin is an inanimate object. It does not have a memory. So does that make the law of averages wrong? If you played the game, you would have only one chance in of winning, but if people played the game, there is a pretty good chance that somebody would win the prize. If people played the game, there would be a good chance that two people might win.
So what does this mean? It means that the probability of winning big in a lottery or on a slot machine is very small, but since there are many, many people playing, somebody will probably win; however, your chances of winning big are very small. A similar situation occurs when couples have children. Suppose a husband and wife have four boys and would like to have a girl. It is incorrect to reason 1 , so it is more likely that the that the chance of having a family of 5 boys is 32 next child will be a girl.
However, after each child is born, the probability that the next child is a girl or a boy for that matter is about The law of averages is not appropriate here. Emery Landon and their 13 boys! Another area where people incorrectly apply the law of averages is in attempting to apply a betting system to gambling games. One such system is doubling your bet when you lose. Consider a game where a coin is tossed. If it lands heads, you win what you bet. If it lands tails, you lose.
If you get tails, you lose one dollar and bet two dollars on the next toss. If you get a tail on the second toss, you bet four dollars on the third toss. If you win, you start over with a one dollar bet, but if you lose, you bet eight dollars on the next toss. With this system, you win every time you get a head. Runs do occur and when they do, hope that they are in your favor. He lost on the twenty-ninth roll. The event took about one hour and twenty minutes.
In in a casino in Puerto Rico at a roulette game, the number 10 occurred six times in succession. There are 38 numbers on a roulette wheel. At a casino in New York in the color red occurred in a roulette game 32 times in a row, and at a casino in Monte Carlo an even number occurred in a roulette game 28 times in a row. So what can be concluded?
First, rare events events with a small probability of occurring can and do occur. Aimed at readers already familiar with applied mathematics at an advanced undergraduate level or higher, it is of interest to scientists concerned with inference from incomplete information Probability For Dummies.
Packed with practical tips and techniques for solving probability problemsIncrease your chances of acing that probability exam -- or winning at the casino! Whether you're hitting the books for a probability or statistics course or hitting the tables at a casino, working out probabilities can be problematic. This book helps you even the odds. It may take up to minutes before you receive it.
The file will be sent to your Kindle account. It may takes up to minutes before you received it. Please note : you need to verify every book you want to send to your Kindle. Check your mailbox for the verification email from Amazon Kindle. Related Booklists. Detailed examples make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce key ideas. It's a no-brainer!
You'll learn about: Classical probability Game theory Actuarial science Addition rules Bayes' theorem Odds and expectation Binomial distribution Simple enough for a beginner, but challenging enough for an advanced student, Probability Demystified , Second Edition, helps you master this essential subject.
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