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.
Accepted Solution
A:
Answer:a) It is binomial distributionb) 0.08224c) 0.05719d) 0.9428e) 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% = 0.85Then the number of flights on a certain route are on time follows a binomial distribution, where[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]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 = 18P(exactly 18 flights are on time)We have to evaluate:[tex]P(x = 18) = \binom{24}{18}(0.85)^{18}(1-0.85)^{6} = 0.0978 + 0.0230 = 0.08224 = 8.2\%[/tex]c) We have to evaluate:P( fewer than 18 flights are on time)[tex]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\%[/tex]d) We have to evaluate:P( at least 18 flights are on time.)[tex]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\%[/tex]e) We have to evaluate:P(between 16 and 18 flights, inclusive, are on time.)[tex]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\%[/tex]