Question

Let N = the number of the tossing a coin when we get the first HEAD. M = the number of the tossing a coin when we get the second HEAD. We assume probability of getting a HEAD is p. Find the Probability Distribution of N and M and are they independent?

Answer 1

We assume probability of getting a HEAD is p

N = the number of the tossing a coin when we get the first HEAD

P[ N = 1 ] = p

P[ N = 2 ] = (1-p)*p ( first toss was not head and second was head )

P[ N = 3 ] = (1-p)^2*p

P[ N = k ] =

M = the number of the tossing a coin when we get the second HEAD

P[ M = 1 ] = 0

P[ M = 2 ] = p*p ( head in both tosses )

P[ M = 3 ] = p*(1-p)*p + (1-p)*p*p = 2*p^2*(1-p)

P[ M = 4 ] = p*(1-p)^2*p + (1-p)*p*(1-p)*p + (1-p)^2*p*p= 3*p^2*(1-p)^2

P[ M = k ] =

Now,

M and N are independent if P[ M | N ] = P[ M ]

P[ M = 2 | N = 1 ] = p

P[ M = 2 ] = p^2

P[ M | N ] /= P[ M ]

Hence, the events are not independent