2021/8/17 HW4 – Intro to Probability

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Intro to Probability – Homework Assignment 4

Please solve the problems and type the solutions up using LateX and the template provided. Naming instructions for the file you will submit are contained in the file itself. Please justify your answers and don’t simply give the answer.

Problem 1 If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is for ?

Problem 2 Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly 3 white balls?

Problem 3 Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second and third cards are spades.

Problem 4 In a certain community, 36% of the families own a dog and 22% of the families that own a dog also own a cat. In addition, 30% of the families own a cat. What is

a. the probability that a randomly selected family owns both a dog and a cat?

b. the conditional probability that a randomly selected family owns a dog given that it owns a cat?

Problem 5 Urn A contains 2 white and 4 red balls, whereas urn B contains 1 white and 1 red ball. A ball is randomly chosen from urn A and put into urn B, and a ball is then randomly selected from urn B. What is

a. the probability that the ball selected from urn B is white?

b. the conditional probability that the transferred ball was white given that a white ball is selected from urn B?

i i = 1, … , 12

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2021/8/17 HW4 – Intro to Probability

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Problem 6 a. A gambler has a fair coin and a two-headed coin in his pocket. He selects one of the coins at random; when he flips it, it shows heads. What is the probability that it is the fair coin?

b. Suppose that he flips the same coin a second time and, again, it shows heads. Now what is the probability that it is the fair coin?

c. Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

Problem 7 Let . Express the following probabilities as simply as possible

Problem 8 A ball is in anyone of boxes and is in the -th box with probability . If the ball is in box , a search of that box will uncover it with probability . Show that the conditional probability that the ball is in box , given that a search of box did not uncover it, is

Problem 9 An event is said to carry negative information about an event iff

This is denoted by . Either give a proof or provide a counterexample for the following assertions

a. If , then .

b. If and , then .

c. If and , then .

Problem 10 The probability of getting a head on a single toss of a coin is . Suppose that A starts and continues to flip the coin until a tail shows up, at which point B starts flipping. Then B continues to flip until a tail comes up, at which point A takes over, and so on. Let

E ⊂ F

P (E|F ), P (E|F c), P (F |E), P (F |E c).

n i Pi i αi

j i

, if j ≠ i, and , if i = j. Pj

1 − αiPi

(1 − αi)Pi 1 − αiPi

E

P (E|F ) ≤ P (E).

F ↘ E

F ↘ E E ↘ F

F ↘ E E ↘ G F ↘ G

F ↘ E G ↘ E F ∩ G ↘ G

p

Pn,m

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denote the probability that A accumulates a total of heads before B accumulates . Show that

n m

Pn,m = p Pn−1,m + (1 − p)(1 − Pm,n).

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