Probability Question: Coin Toss

Question

Suppose you have two coins in a sack, where one is fair and the other is weighted to come up heads only 1/31/3 of the time. Draw one at random, and flip until you get a head. Let X be the number of tosses it takes until you get the first head.

a) What is the probability that it takes you exactly 4 flips to get a head? Do NOT round your answer!

b) What is the probability that it takes you 3 or more flips before you get a head? Do NOT round your answer!

c) What is the expected number of tosses until you get a head, i.e. E[X]?

Answer 1

Since there are 2 coins in the sack that you choose randomly, you pick each coin with probability 1/2. In the case where you pick the fair coin, then the probabilities of a,b,c are:

a) (1/2)^4

Because you need the first 3 coins to be tails, and the 4th one to be heads. Each side occurs with probability 1/2.

b) (1/2)^2

Because you need the first 2 coins to be tails so you can get a head on the 3rd or more try.

c) 2

Because you're expected to get 1 head from every 2 tosses.

Now, there's also the possibility that you pick the unfair coin, in which case the probabilities are:

a) (2/3)^3*(1/3)

Because you want the first 3 coins to be tails, and those come up 2/3 of the time; then you want the 4th to be a head.

b) (2/3)^2

Because again you need the first two flips to be tails to be able to get a head on the 3rd flip and more.

c) 3

Because that's the expected outcome.

Now for each of a,b,c, you have 1/2 chance of picking either coin, so the actual answer is each answer weighted by 1/2:

a) (1/2)^5 + (1/2)*(2/3)^3*(1/3)

b) (1/2)^3 + (2/3)^2*(1/2)

c) 2/2 + 3/2