Find the expected value of 𝑋.

Question

An unfair coin will land “heads up” with probability 0.4 when it is flipped. This coin is to be flipped repeatedly until a total of 2 heads have appeared or until a total of 2 tails have appeared. The random variable 𝑋 is defined to be the total number of flips in the experiment. Find the expected value of 𝑋.

Answer 1

First off, we determine how many tosses we are going to make. And it isn't hard.

We have only two outcomes, head or tail. If we have to stop when 2 head or 2 tails appear we have only six outcomes , two with two tosses and two with three tosses. (order matters because we stop only after the conditions are met. )

1. H. H. (X = two tosses). P = .16

2 T. T. (X = two tosses). P = .36

3. T. H. H (X = three tosses). P = .6x.4x.4 = .096

4 T. H. T. (X = three tosses) P = .144

5. H. T. H. (X = three tosses) P = .096

6. H. T. T. (X = three tosses) P = .144

Expected value is. ∑ P(X_i)·X_i = 3*2*.144 + 3*2*.096 + 2*.36 + 2*.16 = 2.48

easy as 3.1415926