Question

Identify the statements that explain why this is a binomial experiment. Select all that apply. According to an​ airline, flights on a certain route are on time 85​% of the time. Suppose 24 flights are randomly selected and the number of​ on-time flights is recorded. ​(a) Explain why this is a binomial experiment. ​(b) Find and interpret the probability that exactly 18 flights are on time. ​(c) Find and interpret the probability that fewer than 18 flights are on time. ​(d) Find and interpret the probability that at least 18 flights are on time. ​(e) Find and interpret the probability that between 16 and 18 ​flights, inclusive, are on time.

Answer 1

Answer:

a) It is binomial distribution

b) 0.08224

c) 0.05719

d) 0.9428

e) 0.13344    

Step-by-step explanation:

We are given the following information:

a) We treat flights on a certain route are on time as a success.

P(flights on a certain route are on time ) = 85% = 0.85

Then the number of flights on a certain route are on time follows a binomial distribution, where

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

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

b) Now, we are given n = 24 and x = 18

P(exactly 18 flights are on time)

We have to evaluate:

$P(x = 18) = \binom{24}{18}(0.85)^{18}(1-0.85)^{6} = 0.0978 + 0.0230 = 0.08224 = 8.2\%$

c) We have to evaluate:

P( fewer than 18 flights are on time)

$P(x < 18) =\binom{24}{18}(0.85)^{0}(1-0.85)^{24} +...+ \binom{24}{18}(0.85)^{17}(1-0.85)^{7} = 0.05719= 5.7\%$

d) We have to evaluate:

P( at least 18 flights are on time.)

$P(x \geq 18) =\binom{24}{18}(0.85)^{18}(1-0.85)^{6} +...+ \binom{24}{18}(0.85)^{24}(1-0.85)^{0} = 0.9428= 94.28\%$

e)  We have to evaluate:

P(between 16 and 18 ​flights, inclusive, are on time.)

$P(16 \leq x \leq 18) =\binom{24}{18}(0.85)^{18}(1-0.85)^{6} +\binom{24}{17}(0.85)^{17}(1-0.85)^{7}+ \binom{24}{16}(0.85)^{16}(1-0.85)^{8} =0.08224+ 0.0373+0.0139=0.13344= 13.34\%$