Answer :

Given that, a person buys a lottery ticket in 50 lotteries.


The probability of winning a prize is 1/100.


Let p be the probability of winning a prize.


Then,


And q be the probability of not winning a prize.


We can write,


p + q = 1


q = 1 – p





Let X be a random variable representing the number of times the person wins the lottery out of n lotteries.


Then, the probability of the person winning the lottery r times out of n times is given by this Binomial distribution.


P (X = r) = nCrprqn-r


Here, n = 50


So, putting the value of n, p, and q in the formula of P (X = r), we get


…(A)


(i). We need to find the probability that he will win the prize at least once.


The probability is given by,


Probability = P (X ≥ 1)


Or this can be written as,


P (X ≥ 1) = 1 – P (X < 1)


P (X ≥ 1) = 1 – P (X = 0) …(B)


So, put r = 0 in equation (A) and then substitute in equation (B), we get






Thus, the probability that he will win the prize at least once is .


(ii). We need to find the probability that he will win the prize exactly once.


The probability is given by,


Probability = P (X = 1)


Put r = 1 in equation (A), we get







Thus, the probability that he will win the prize exactly once is .


(iii). We need to find the probability that he will win at least twice.


The probability is given by,


Probability = P (X ≥ 2)


Or can be expressed as,


P (X ≥ 2) = 1 – P (X < 2)


P (X ≥ 2) = 1 – [P (X = 0) + P (X = 1)] …(B)


Put r = 0 and r = 1 in equation (A) one by one and then substitute it in the equation (B), we get









Thus, the probability that he will win the prize at least twice is .


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