Question

Let x be a binomial random variable with n = 100 and p = 0.2. Find approximations to these probabilities. (Round your answers to four decimal places.) (a) P(x > 22) = Incorrect: Your answer is incorrect. (b) P(x ≥ 22) = 0.3085 Incorrect: Your answer is incorrect. (c) P(18 < x < 28) = 0.4813 Incorrect: Your answer is incorrect.

Answer 1

a) 0.2611

b) 0.6038

Explanation:

We are given the following information:

Let x be a binomial random variable with n = 100 and p = 0.2.

P(Success) = 0.2

Formula:

`P(X=x) = \binom{n}{x}.p^x.(1-p)^(n-x)`

where n is the total number of observations, x is the number of success, p is the probability of success.

a) Now, we are given n = 100 and x = 22

We have to evaluate:

`\bold{P(x > 22)} = P(x = 23) +...+ P(x = 100) \\= \binom{100}{23}(0.2)^(23)(1-0.2)^(100-23) +...+ \binom{100}{100}(0.2)^(100)(1-0.2)^0\\= 0.0719 +...+ 0.000001\\=0.2611`

b) Now, we are given n = 100 and 18 < x < 28

We have to evaluate:

`\bold{P(18 < x < 28)} = P(x \leq 28) - P(x \leq 18)\\= P(x = 19) +...+ P(x = 27) \\= \binom{100}{19}(0.2)^(19)(1-0.2)^(100-19) +...+ \binom{100}{27}(0.2)^(27)(1-0.2)^(100-27)\\= 0.09807 +...+ 0.0216 = 0.6038`