A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is 1/100. What is the probability that he will win a prize

i. at least once

ii. exactly once

iii. at least twice?

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) = ^{n}C_{r}p^{r}q^{n-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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