5.1 Sets, Counting, and Venn Diagrams
reviewcard
CHAPTER 5 PR OBABI L I TY
Section Summaries
Vocabulary Set, 178
5.1 Sets, Counting, and Venn Diagrams (pp. 178–186)
Universal set, 178
c
For a set A, its complement A is the set of all elements not in A. For sets A and B, their intersection A d B is the set of elements that are in both A and B, and their union A h B is the set of elements that are in either A or B or both. In the Venn diagrams below, U stands for the universal set of all elements being considered. U
A
A
B
U
B U
A
Intersection, 179 Disjoint sets, 179 Union, 180 Complement, 180 Empty set, 180 Subset, 180 Complementary principle, 180
Ac
A艛B
A傽B
Addition principle, 181
For a finite set A, n(A) means the number of elements in A. The following formulas enable us to find the number of elements in one set by counting the elements in other sets: n 1 Ac 2 5 n 1 U 2 2 n 1 A 2 n1A h B2 5 n1A2 1 n1B2 2 n1A d B2
Complementary principle of counting Addition principle of counting
Tree diagram, 182 Multiplication principle, 182 Number of subsets, 184 Factorial, 186 Permutations, 188 Combinations, 190 Sample space, 192 Events, 193
The multiplication principle says that if two choices are to be made and one can be made in m ways and the other can be made in n ways, and if the ways can be freely combined, then the two choices together can be made in m # n ways. The multiplication principle generalizes to more than two choices. A set with n members has 2n subsets.
Equally-likely outcomes, 194 Probability space, 197 Addition rule, 197 Disjoint events, 198 Mutually exclusive events, 198
5.2 Permutations and Combinations (pp. 186–192)
Conditional probability, 201
Permutations are ordered choices; combinations are choices without regard to order. That is, in permutations a different order means a different object (abc and bca are different), but in combinations changing the order does not represent a new object (abc and bca are counted as the same). If order matters (as with letters in words or listings of people for president, vice president, and treasurer), use permutations; if order does not matter (as with committees or hands of cards), use combinations.
Product rule, 203
r factors 7666648466669 # # # # # # 1n 2 r 1 12 nPr 5 n 1 n 2 1 2 n # 1n 2 12 # # # # # 1n 2 r 1 12 n! 5 nCr 5 r # 1r 2 12 # # # # # 1 r! 1 n 2 r 2 !
Independent events, 206 Dependent events, 206 Bayes’ formula, 210 Random variable, 214 Probability distribution, 214 Expected value, 215
Permutations (orderings)
Mean, 215
Combinations (collections)
Binomial random variable, 218
Binomial distribution, 218
5.3 Probability Spaces (pp. 192–200) For a random experiment, the sample space is the set S of possible outcomes. We assign to each possible outcome a probability between 0 and 1 such that the sum of the probabilities is 1. If the n possible events are equally likely, we assign each probability 1/n. A probability space is a sample space S together with an assignment of probabilities.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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An event E is a subset of S, and the probability of the event E is Adding the probabilities for P 1 E 2 5 a P 1 ei 2 each outcome in the event All ei in E In particular, P 1 S 2 5 1, P 1 [ 2 5 0, and P 1 E c 2 5 1 2 P 1 E 2 . The probability of a union A h B of two events is found by the addition rule: P1A h B2 5 P1A2 1 P1B2 2 P1A d B2 If events A and B are mutually exclusive or disjoint (that is, A d B 5 [ 2 , this rule simplifies to P1A h B2 5 P1A2 1 P1B2
5.4 Conditional Probability and
Independence (pp. 201–208) For events A and B, the conditional probability of A given B is P1A d B2 P 1 A given B 2 5 For P(B) 0 P1B2 P 1 A given B 2 can also be found by restricting P 1 A 2 to the sample space where B occurs. The above formula can be rewritten as: Product Rule for P 1 A d B 2 5 P 1 A 2 # P 1 B given A 2 Probability This formula is useful in calculating probabilities along branches in tree diagrams. Two events A and B are independent if P1A d B2 5 P1A2 # P1B2
5.6 Random Variables and
Distributions (pp. 213–221) A random variable X is an assignment of a number to each outcome in a sample space. The probability distribution of X is the collection of probabilities P 1 X 5 x 2 for each possible value x of X. The expected value or mean of X is E 1 X 2 5 x1 # P 1 X 5 x1 2 1 # # # 1 xn # P 1 X 5 xn 2 Sometimes denoted m Bernoulli trials are independent repetitions of an experiment that results in success with probability p (and therefore failure with probability 1 2 p 2 . The number X of successes in n Bernoulli trials is called a binomial random variable with parameters n and p and has probability distribution and mean P 1 X 5 k 2 5 nCk pk 1 1 2 p 2 n2k
For k 5 0, 1, ... , n
E1X2 5 n # p
The following graphs show the binomial distribution for n 5 6 and different values of p. Notice that the second graph is symmetric and that values away from p 5 12 skew the graph to one side or the other.
0
1
2
3
4
5
6
4
5
6
4
5
6
1 p= 3
Probabilities multiply
Be careful! When do you add probabilities and when do you multiply them? You add when the events are disjoint and you want one or the other to occur (subtracting the intersection if the events are not disjoint); you multiply if the events are independent and you want both to occur (using conditional probability if the events are not independent).
5.5 Bayes’ Formula (pp. 209–213) For pairwise disjoint sets S1, S2, ... , Sn whose union is the entire sample space S1 h S2 h # # # h Sn 5 S, Bayes’ formula reverses the order in conditional probability: P 1 S1 given A 2 5
P 1 S1 2 # P 1 A given S1 2 P 1 S1 2 # P 1 A given S1 2 1 # # # 1 P 1 Sn 2 # P 1 A given Sn 2
0
1
2
3 1 p= 2
0
1
2
3 2 p= 3
Under each graph the mean (calculated from E 1 X 2 5 n # p 2 is shown in a triangle to indicate that the distribution “balances” at that point.
In practice, we carry out this calculation using a tree diagram, where the numerator is the product along one branch and the denominator is the sum of several branches, using just the definition of conditional probability.
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Practice Test
5.1 Sets, Counting, and Venn Diagrams 1. Venn Diagrams For the Venn diagram below, find: a. n 1 A 2
b. n 1 A h B 2
c. n 1 B c 2
A
d. n 1 Ac d B 2 U
B 12
5
15 9
2. Purchases A survey of 1000 homeowners found that during the last year 230 bought an automobile, 340 bought a major appliance, and 540 bought neither. How many homeowners had bought both an automobile and a major appliance?
5.2 Permutations and Combinations 3. Committees How many 4-member committees can be formed from a club consisting of 20 students? What if the committee is to consist of a president, vice-president, secretary, and treasurer?
5.3 Probability Spaces 4. Marbles A box contains three marbles: one blue, one yellow, and one red. A first marble is chosen, then it is replaced and a second marble is chosen. Find the sample space. Then find the sample space if the first marble is not replaced before the second is chosen.
5. Smoking and Weight For a randomly selected person, the probability of being a smoker is 0.35, the probability of being overweight is 0.40, and the probability of being both a smoker and overweight is 0.20. What is the probability of being neither a smoker nor overweight?
6. Dice If you roll one die, find the probability of: a. Rolling at most 2. b. Rolling an even number.
7. Lottery In a lottery you choose 4 numbers out of 40. Then 5 numbers are announced, and you win something if you have 4 of the 5 numbers. What is the probability that you win something?
8. Defective Products A store shelf has 50
9. Committees A committee of 3 is to be formed from your class of 30. What is the probability that both you and your best friend will be on it?
5.4 Conditional Probability and Independence 10. Dice If you roll two dice, what is the probability of at least one six given that the sum of the numbers is seven?
11. Coins For the experiment of tossing a coin twice,
with sample space 5 1 H, H 2 , 1 H, T 2 , 1 T, H 2 , 1 T, T 2 6 , are the following events independent or dependent? a. At least one head and at most one head. b. Heads on first toss and same face on both tosses.
12. Coins Five coins are tossed. Find the probability of getting a. All heads. b. All the same outcome. c. Alternating outcomes.
5.5 Bayes’ Formula 13. Manufacturing Defects A soft-drink bottler has three bottling machines: A, B, and C. Machine A bottles 25% of the company’s output, machine B bottles 35%, and machine C bottles 40%. It is known that 4% of the bottles produced by machine A are defective, as are 3% of the bottles produced by machine B and 2% of the bottles from machine C. If a randomly selected bottle from the factory’s output is found to be defective, what is the probability that it was produced by machine A?
5.6 Random Variables and Distributions 14. Coins You toss two coins. You win $34 if you get double heads, and otherwise you lose $2. Find and graph the probability distribution of your winnings.
15. Marbles Two marbles are chosen at random and without replacement from a box containing 5 red and 4 green marbles. If X is the number of green marbles chosen, find and graph the probability distribution of X.
lightbulbs, of which 2 are defective. If you buy 4, what is the probability that you get both defective bulbs? Neither defective bulb?
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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16. Raffle Tickets Five hundred raffle tickets are sold for the following prizes: one first prize worth $1000, two second prizes worth $150, and 10 third prizes worth $25. Find the expected value of a ticket.
18. Product Quality Of the products a company produces, 1% have hidden defects. If you buy a dozen of them, what is the mean of the number of defective ones in your purchase?
17. Sales A television salesperson estimates that a customer will buy a $900 TV with probability 101 , a $500 TV with probability 12, a $200 TV with probability 101 , and otherwise buy nothing. What is the expected value of the sale?
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CHAPTER 5 PR OBABI L I TY
Section Summaries
Vocabulary Set, 178
5.1 Sets, Counting, and Venn Diagrams (pp. 178–186)
Universal set, 178
c
For a set A, its complement A is the set of all elements not in A. For sets A and B, their intersection A d B is the set of elements that are in both A and B, and their union A h B is the set of elements that are in either A or B or both. In the Venn diagrams below, U stands for the universal set of all elements being considered. U
A
A
B
U
B U
A
Intersection, 179 Disjoint sets, 179 Union, 180 Complement, 180 Empty set, 180 Subset, 180 Complementary principle, 180
Ac
A艛B
A傽B
Addition principle, 181
For a finite set A, n(A) means the number of elements in A. The following formulas enable us to find the number of elements in one set by counting the elements in other sets: n 1 Ac 2 5 n 1 U 2 2 n 1 A 2 n1A h B2 5 n1A2 1 n1B2 2 n1A d B2
Complementary principle of counting Addition principle of counting
Tree diagram, 182 Multiplication principle, 182 Number of subsets, 184 Factorial, 186 Permutations, 188 Combinations, 190 Sample space, 192 Events, 193
The multiplication principle says that if two choices are to be made and one can be made in m ways and the other can be made in n ways, and if the ways can be freely combined, then the two choices together can be made in m # n ways. The multiplication principle generalizes to more than two choices. A set with n members has 2n subsets.
Equally-likely outcomes, 194 Probability space, 197 Addition rule, 197 Disjoint events, 198 Mutually exclusive events, 198
5.2 Permutations and Combinations (pp. 186–192)
Conditional probability, 201
Permutations are ordered choices; combinations are choices without regard to order. That is, in permutations a different order means a different object (abc and bca are different), but in combinations changing the order does not represent a new object (abc and bca are counted as the same). If order matters (as with letters in words or listings of people for president, vice president, and treasurer), use permutations; if order does not matter (as with committees or hands of cards), use combinations.
Product rule, 203
r factors 7666648466669 # # # # # # 1n 2 r 1 12 nPr 5 n 1 n 2 1 2 n # 1n 2 12 # # # # # 1n 2 r 1 12 n! 5 nCr 5 r # 1r 2 12 # # # # # 1 r! 1 n 2 r 2 !
Independent events, 206 Dependent events, 206 Bayes’ formula, 210 Random variable, 214 Probability distribution, 214 Expected value, 215
Permutations (orderings)
Mean, 215
Combinations (collections)
Binomial random variable, 218
Binomial distribution, 218
5.3 Probability Spaces (pp. 192–200) For a random experiment, the sample space is the set S of possible outcomes. We assign to each possible outcome a probability between 0 and 1 such that the sum of the probabilities is 1. If the n possible events are equally likely, we assign each probability 1/n. A probability space is a sample space S together with an assignment of probabilities.
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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C h a p t e r 3 : P l a n n i n g a n d D e cCi h s iaopnt eMr a5k: i nP g robability
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An event E is a subset of S, and the probability of the event E is Adding the probabilities for P 1 E 2 5 a P 1 ei 2 each outcome in the event All ei in E In particular, P 1 S 2 5 1, P 1 [ 2 5 0, and P 1 E c 2 5 1 2 P 1 E 2 . The probability of a union A h B of two events is found by the addition rule: P1A h B2 5 P1A2 1 P1B2 2 P1A d B2 If events A and B are mutually exclusive or disjoint (that is, A d B 5 [ 2 , this rule simplifies to P1A h B2 5 P1A2 1 P1B2
5.4 Conditional Probability and
Independence (pp. 201–208) For events A and B, the conditional probability of A given B is P1A d B2 P 1 A given B 2 5 For P(B) 0 P1B2 P 1 A given B 2 can also be found by restricting P 1 A 2 to the sample space where B occurs. The above formula can be rewritten as: Product Rule for P 1 A d B 2 5 P 1 A 2 # P 1 B given A 2 Probability This formula is useful in calculating probabilities along branches in tree diagrams. Two events A and B are independent if P1A d B2 5 P1A2 # P1B2
5.6 Random Variables and
Distributions (pp. 213–221) A random variable X is an assignment of a number to each outcome in a sample space. The probability distribution of X is the collection of probabilities P 1 X 5 x 2 for each possible value x of X. The expected value or mean of X is E 1 X 2 5 x1 # P 1 X 5 x1 2 1 # # # 1 xn # P 1 X 5 xn 2 Sometimes denoted m Bernoulli trials are independent repetitions of an experiment that results in success with probability p (and therefore failure with probability 1 2 p 2 . The number X of successes in n Bernoulli trials is called a binomial random variable with parameters n and p and has probability distribution and mean P 1 X 5 k 2 5 nCk pk 1 1 2 p 2 n2k
For k 5 0, 1, ... , n
E1X2 5 n # p
The following graphs show the binomial distribution for n 5 6 and different values of p. Notice that the second graph is symmetric and that values away from p 5 12 skew the graph to one side or the other.
0
1
2
3
4
5
6
4
5
6
4
5
6
1 p= 3
Probabilities multiply
Be careful! When do you add probabilities and when do you multiply them? You add when the events are disjoint and you want one or the other to occur (subtracting the intersection if the events are not disjoint); you multiply if the events are independent and you want both to occur (using conditional probability if the events are not independent).
5.5 Bayes’ Formula (pp. 209–213) For pairwise disjoint sets S1, S2, ... , Sn whose union is the entire sample space S1 h S2 h # # # h Sn 5 S, Bayes’ formula reverses the order in conditional probability: P 1 S1 given A 2 5
P 1 S1 2 # P 1 A given S1 2 P 1 S1 2 # P 1 A given S1 2 1 # # # 1 P 1 Sn 2 # P 1 A given Sn 2
0
1
2
3 1 p= 2
0
1
2
3 2 p= 3
Under each graph the mean (calculated from E 1 X 2 5 n # p 2 is shown in a triangle to indicate that the distribution “balances” at that point.
In practice, we carry out this calculation using a tree diagram, where the numerator is the product along one branch and the denominator is the sum of several branches, using just the definition of conditional probability.
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Practice Test
5.1 Sets, Counting, and Venn Diagrams 1. Venn Diagrams For the Venn diagram below, find: a. n 1 A 2
b. n 1 A h B 2
c. n 1 B c 2
A
d. n 1 Ac d B 2 U
B 12
5
15 9
2. Purchases A survey of 1000 homeowners found that during the last year 230 bought an automobile, 340 bought a major appliance, and 540 bought neither. How many homeowners had bought both an automobile and a major appliance?
5.2 Permutations and Combinations 3. Committees How many 4-member committees can be formed from a club consisting of 20 students? What if the committee is to consist of a president, vice-president, secretary, and treasurer?
5.3 Probability Spaces 4. Marbles A box contains three marbles: one blue, one yellow, and one red. A first marble is chosen, then it is replaced and a second marble is chosen. Find the sample space. Then find the sample space if the first marble is not replaced before the second is chosen.
5. Smoking and Weight For a randomly selected person, the probability of being a smoker is 0.35, the probability of being overweight is 0.40, and the probability of being both a smoker and overweight is 0.20. What is the probability of being neither a smoker nor overweight?
6. Dice If you roll one die, find the probability of: a. Rolling at most 2. b. Rolling an even number.
7. Lottery In a lottery you choose 4 numbers out of 40. Then 5 numbers are announced, and you win something if you have 4 of the 5 numbers. What is the probability that you win something?
8. Defective Products A store shelf has 50
9. Committees A committee of 3 is to be formed from your class of 30. What is the probability that both you and your best friend will be on it?
5.4 Conditional Probability and Independence 10. Dice If you roll two dice, what is the probability of at least one six given that the sum of the numbers is seven?
11. Coins For the experiment of tossing a coin twice,
with sample space 5 1 H, H 2 , 1 H, T 2 , 1 T, H 2 , 1 T, T 2 6 , are the following events independent or dependent? a. At least one head and at most one head. b. Heads on first toss and same face on both tosses.
12. Coins Five coins are tossed. Find the probability of getting a. All heads. b. All the same outcome. c. Alternating outcomes.
5.5 Bayes’ Formula 13. Manufacturing Defects A soft-drink bottler has three bottling machines: A, B, and C. Machine A bottles 25% of the company’s output, machine B bottles 35%, and machine C bottles 40%. It is known that 4% of the bottles produced by machine A are defective, as are 3% of the bottles produced by machine B and 2% of the bottles from machine C. If a randomly selected bottle from the factory’s output is found to be defective, what is the probability that it was produced by machine A?
5.6 Random Variables and Distributions 14. Coins You toss two coins. You win $34 if you get double heads, and otherwise you lose $2. Find and graph the probability distribution of your winnings.
15. Marbles Two marbles are chosen at random and without replacement from a box containing 5 red and 4 green marbles. If X is the number of green marbles chosen, find and graph the probability distribution of X.
lightbulbs, of which 2 are defective. If you buy 4, what is the probability that you get both defective bulbs? Neither defective bulb?
© 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
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16. Raffle Tickets Five hundred raffle tickets are sold for the following prizes: one first prize worth $1000, two second prizes worth $150, and 10 third prizes worth $25. Find the expected value of a ticket.
18. Product Quality Of the products a company produces, 1% have hidden defects. If you buy a dozen of them, what is the mean of the number of defective ones in your purchase?
17. Sales A television salesperson estimates that a customer will buy a $900 TV with probability 101 , a $500 TV with probability 12, a $200 TV with probability 101 , and otherwise buy nothing. What is the expected value of the sale?
Go to CourseMate for FINITE for additional practice, printable flashcards, and more! Access at login.cengagebrain.com.
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