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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) = 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 .