Questions: According to an airline, flights on a certain route are on time 85% of the time. Suppose 25 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 17 flights are on time. (c) Find and interpret the probability that fewer than 17 flights are on time. (d) Find and interpret the probability that at least 17 flights are on time. (e) Find and interpret the probability that between 15 and 17 flights, inclusive, are on time.

According to an airline, flights on a certain route are on time 85% of the time. Suppose 25 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 17 flights are on time. (c) Find and interpret the probability that fewer than 17 flights are on time. (d) Find and interpret the probability that at least 17 flights are on time. (e) Find and interpret the probability that between 15 and 17 flights, inclusive, are on time.
Transcript text: According to an airline, flights on a certain route are on time $85 \%$ of the time. Suppose 25 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 17 flights are on time. (c) Find and interpret the probability that fewer than 17 flights are on time. (d) Find and interpret the probability that at least 17 flights are on time. (e) Find and interpret the probability that between 15 and 17 flights, inclusive, are on time.
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Solution

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Solution Steps

Step 1: Calculating the probability of exactly 17 successes

Using the formula $P(X = 17) = \binom{n}{k} p^k (1-p)^{n-k}$, where $\binom{n}{k}$ is the binomial coefficient.

Step 2: Substituting the values $n=25$, $p=0.85$, and $k=17$ into the formula

Final Answer:

$P(X = 17) = 0.0175$

Step 1: Calculating the probability of fewer than 17 successes

Using the formula $P(X < 17) = \sum_{i=0}^{k-1} \binom{n}{i} p^i (1-p)^{n-i}$

Step 2: Substituting the values $n=25$, $p=0.85$, and summing from $i=0$ to $k-1$
Final Answer:

$P(X < 17) = 0.008$

Step 1: Calculating the probability of at least 17 successes

Using the formula $P(X \geq 17) = \sum_{i=17}^{n} \binom{n}{i} p^i (1-p)^{n-i}$

Step 2: Substituting the values $n=25$, $p=0.85$, and starting from $k=17$ to $n=25$
Final Answer:

$P(X \geq 17) = 0.992$

Step 1: Calculating the probability of having between 15 and 17 successes, inclusive

Using the formula $P(15 \leq X \leq 17) = \sum_{i=15}^{b} \binom{n}{i} p^i (1-p)^{n-i}$

Step 2: Substituting the values $n=25$, $p=0.85$, and summing from $i=15$ to $b=17$
Final Answer:

$P(15 \leq X \leq 17) = 0.025$

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