Question 4: 30% of people caught the flu this year. 80% of people got the flu shot. 20% of the people who got the flu shot still caught the flu.¹ a) Use the definition of conditional probability and any result we've seen in class to prove: Pr(A) = Pr(A|B) · Pr(B) + Pr(A|B) · Pr(B) . b) What is your probability of catching the flu if you did not get the flu shot? c) Use the definition of conditional probability to prove Bayes' Theorem: Pr(A|B) = Pr(B|A). Pr(A) Pr(B) d) You know someone who caught the flu. What is the probability that they got the flu shot?
Question 4: 30% of people caught the flu this year. 80% of people got the flu shot. 20% of the people who got the flu shot still caught the flu.¹ a) Use the definition of conditional probability and any result we've seen in class to prove: Pr(A) = Pr(A|B) · Pr(B) + Pr(A|B) · Pr(B) . b) What is your probability of catching the flu if you did not get the flu shot? c) Use the definition of conditional probability to prove Bayes' Theorem: Pr(A|B) = Pr(B|A). Pr(A) Pr(B) d) You know someone who caught the flu. What is the probability that they got the flu shot?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
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Chapter1: Combinatorial Analysis
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Answer both Q4, Q5
![Question 4: 30% of people caught the flu this year. 80% of people got the flu shot. 20% of
the people who got the flu shot still caught the flu.¹
a) Use the definition of conditional probability and any result we've seen in class to prove:
Pr(A) = Pr(A|B) · Pr(B) + Pr(A|B) · Pr(B)
b) What is your probability of catching the flu if you did not get the flu shot?
c) Use the definition of conditional probability to prove Bayes' Theorem:
Pr(A|B) =
Pr(B|A) · Pr(A)
Pr(B)
d) You know someone who caught the flu. What is the probability that they got the flu
shot?
Question 5: You are playing 5 card poker with your friends. In this game you are initially
dealt 5 cards and on each round of betting you can exchange any number of cards to try and
get a better hand. You are using a standard deck of 52 cards, and any cards you exchange
go into a discard pile and are no longer in play for this hand. All cards are dealt uniformly
at random.
In your first hand you are dealt {5◇, 50, 7◊, 8♣, A◊}. You keep the pair of 5's and the
ace and exchange the 7 and 8 for two more cards. Let C be the event that you receive at
least one more ace, and let D be the event that you receive at least one more 5.
a) What is Pr(Cn D)?
b) What is Pr(CUD)?
c) Are events C and D independent? In other words, is Pr(C^ D) = Pr(C) · Pr(D)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F237ed487-4951-49d2-9402-d31d2e9fdf41%2F761896c4-65da-4d21-8e55-a0f4eb7f258d%2F42soxhk_processed.png&w=3840&q=75)
Transcribed Image Text:Question 4: 30% of people caught the flu this year. 80% of people got the flu shot. 20% of
the people who got the flu shot still caught the flu.¹
a) Use the definition of conditional probability and any result we've seen in class to prove:
Pr(A) = Pr(A|B) · Pr(B) + Pr(A|B) · Pr(B)
b) What is your probability of catching the flu if you did not get the flu shot?
c) Use the definition of conditional probability to prove Bayes' Theorem:
Pr(A|B) =
Pr(B|A) · Pr(A)
Pr(B)
d) You know someone who caught the flu. What is the probability that they got the flu
shot?
Question 5: You are playing 5 card poker with your friends. In this game you are initially
dealt 5 cards and on each round of betting you can exchange any number of cards to try and
get a better hand. You are using a standard deck of 52 cards, and any cards you exchange
go into a discard pile and are no longer in play for this hand. All cards are dealt uniformly
at random.
In your first hand you are dealt {5◇, 50, 7◊, 8♣, A◊}. You keep the pair of 5's and the
ace and exchange the 7 and 8 for two more cards. Let C be the event that you receive at
least one more ace, and let D be the event that you receive at least one more 5.
a) What is Pr(Cn D)?
b) What is Pr(CUD)?
c) Are events C and D independent? In other words, is Pr(C^ D) = Pr(C) · Pr(D)?
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