Question

Let Z be the number of the flipping when first Head appears, and M be the number of flipping when second head appears. We know the probability of getting a head is p and probability of getting a tail is then 1-p.

The questions are: a) Find the probability distribution of Z and M. b) Are Z and M independent? Give the reason

Answer 1

Z is the number of flippings when first head appears. We know that the probability of getting a head is p.

So the probability of getting head in the first flip is p

The probability of getting the first head in the second flip is (1-p)p.

Therefore, the probability distribution of Z is given as:

P(Z=k) = p*(1-p)^(k-1) which is nothing but the geometric distribution.

M is the number of flippings when the second head appears.

Let us consider that M denote the number over and above Z taken by an individual to get the second head.

In such a case the probability that second head comes in the next turn is:

P(M=1) = p

Similarly the probability distribution function is given as:

P(M=k) = p*(1-p)^(k-1)

which is nothing but the geometric distribution.

Also we have:

P(Z=z,M=m) = p*p*(1-p)^(z+m-2)

whcih is nothing but the product of the individual probabilities.

Hence Z and M are independent.