1. (2.2) A fair coin is flipped three times (the flips are independent). a. What

Question

1.      (2.2) A fair coin is flipped three times (the flips are independent).

a.       What is the probability the first flip is a head given the second flip is a head?

b.      What is the probability the first flip is a head given there is at least one head in the three flips?

c.       What is the probability the first flip is a head given there are at least two heads in the three flips?

2.      (2.3) a coin with probability p of coming up heads is flipped three times (the flips are independent).

a.       What is the probability of getting at least two heads given there is at least one head?

b.      What is the probability of getting at least one head given there are at least two heads?

Explanation / Answer

Question 1:

a) The result of the first toss does not depend on the second toss. Therefore the probability that the first flip is a head given that the second flip is a head is simply the probability that the first flip is a head which would be equal to 0.5 for a fair coin.

Therefore 0.5 is the required probability here.

b) Probability that the first flip is a head given that there is at least one head in the three flips is computed as: ( Using Bayes theorem )

= ( Probability that there is a head in the first flip ) / Probability that there is at least one head in the three flips

Now the probability that there is at least one head in the three flips is computed as:

= 1 - Probability that there is no head in three flips.

Now there are a total of 8 possibilities in 3 coin flips. These are: HHH, HHT, HTH, THH, TTH, THT, HTT and TTT. Out of these 8 possibilities, there is only 1 possibility with all 3 tails. Therefore the probability to have at least one head is computed as

= 1- (1/8) = 0.875

Therefore, the probability the first flip is a head given there is at least one head in the three flips is computed as:

= ( Probability that there is a head in the first flip ) / Probability that there is at least one head in the three flips

= 0.5 / 0.875

= 0.5714

Therefore 0.5714 is the required probability here.

c) Now here we will first compute the probability that there are at least 2 heads. The cases with at least 2 heads are: HHT, HTH, THH and HHH. Therefore the probability of at least 2 heads is computed as:

= 4/8 = 0.5

Now the probability that there is head on the first place and there are at least 2 heads is computed as: 3/8 = 0.375 because there are only 3 cases: HHT, HHH and HTH.

Now Using Bayes theorem, the probability the first flip is a head given there are at least two heads in the three flips is computed as:

= probability that there is head on the first place and there are at least 2 heads / Probability that there are atleast 2 heads

= 0.375 / 0.5

= 0.75

Therefore 0.75 is the required probability here.

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