(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm currently trying to revise for exams and really struggling on this problem:

Suppose you have 3 coins that look identical (ie don't know which is which) with probabilites of 1/4, 1/2 and 3/4 of showing a head.

1. If you pick a coin at random and flip it, what is the probability of a head?

2. If you flip all 3 coins 1 after the other what is the probability of getting 3 heads? Are the three events Aj that the jth coin shows a head for j=1,2,3 independent?

3. You pick a coin at random and flip it twice and get a head both times. What is the conditional probability that you picked the coin with a probability of 3/4 of getting a head?

4. You pick one of the other 2 coins at random and flip it. What is the probability of a head?

2. Relevant equations

Partition Theorem P(A) = [tex]\sum[/tex] P(A | Bi)P(Bi)

P(B|A) = P(A|B)P(B)/P(A)

3. The attempt at a solution

1. I think this is 1/2, by adding up the probabilities and dividing by 3.

2. Would I get the probability of all heads by doing 1/4 x 1/2 x 3/4 = 3/32? As I assume that getting a head on one coin does not affect the other coins?

But then I'm not sure about the next part of (2). Is P(A1) = 1/2 by (1). The I tried to use the partition theorem for P(A2)

P(A2) = P(A2|A1)P(A1) + P(A2|not A1) P(not A1)

P(A1) = 1/2 = P(not A1) but apart from this I don't know.

3. P(3/4 coin | 2 heads) = P (2 heads| 3/4 coin)P(3/4 coin) / P(2 heads) = 3/4 x 3/4 x 1/3 / P(2 heads), but I don't know P(2 heads). Would drawing a probability tree work to find this?

4. I have not yet attempted this as I wanted to be clear on the rest of the question 1st.

Thanks :)

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# Homework Help: Conditional probability and partitions

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