Question

A coin with probability p>0 of turning up heads is tossed 4 times. Let X be the number of times heads are tossed.

(a) Find the probability function of X in terms of p.

(b) The result above can be extended to the case of n independent tosses (that is, for a generic number of tosses), and the probability function in this case receives a very specific name. Find the name of this particular probability function.

Notice that the probability of turning up tails in one toss is 1−p.

Answer 1

Probability of turning up heads in one toss = p >0

Probability of turning up tails in one toss = 1−p

number of tosses = 4

Let X be the number of times heads are tossed

(a) Find the probability function of X in terms of p.

Since, the outcome of each toss is independent of other and the probability of turning up heads is same in each case.

P[ getting k heads ] = getting k heads with probability p * getting ( 4 - k ) tails with probability ( 1 - p ) * arranging k heads out of 4 places

getting k heads with probability p = p^k

getting ( n - k ) tails with probability ( 1 - p ) = (1 - p)^(4 - k)

arranging k heads out of 4 places = 4Ck

P[ getting k heads ] = P[ X = k ] = 4Ck*p^k*(1 - p)^(4 - k)

(b) The result above can be extended to the case of n independent tosses

Replacing 4 by n

P[ X = k ] = nCk*p^k*(1 - p)^(n - k)

This is binomial probability distribution used when the events are independent and the probability of outcome is same at each trial