Question

A $1 coin is tossed until a head appears, and let N be the total number of times that the $1 coin is tossed. A $5 coin is then tossed N times. Let X count the number of heads appearing on the tosses of the $5 coin. Determine P(X = 0) and P(X = 1).

Answer 1

Here, N is the total number of tosses till a head appears. Thus the probabilities here are computed as:
P(N = 1) = 0.5
P(N = 2) = 0.5*0.5 = 0.5²
and so on.. P(N = n) = 0.5ⁿ

The probability here is computed as: (using law of total probability )
P(X = 0) = P(N = 1)P(X = 0 | N = 1) + P(N = 2)P(X = 0 | N = 2) + ......

Using the sum of an infinite GP sum, we get here:

Therefore 1/3 is the required probability here.

Now P(X = 1) is computed in same way here as:

P(X = 1) = P(N = 1)P(X = 1 | N = 1) + P(N = 2)P(X = 1 | N = 2) + ......

Multiplying both sides by 0.5², we get here:

Subtracting the last equation from the second last one, we get here:

Using the sum of an infinite GP sum, we get here:

Therefore 0.4444 is the required probability here.