At a particular airport, 81% of the flights arrived on time last year. If 15 flights are randomly selected, find the probability that 11 or fewer flights are on time.
The correct answer and explanation is :
To solve this problem, we are looking for the probability that 11 or fewer flights out of 15 randomly selected flights arrive on time, given that 81% of the flights arrived on time last year. This is a binomial probability problem because there are two possible outcomes for each flight (either it is on time or it is not), and the probability of success (a flight being on time) remains constant for each trial.
Step 1: Define the parameters of the binomial distribution.
In this case:
- n = 15 (the number of trials, or flights selected),
- p = 0.81 (the probability of a flight being on time),
- x = the number of successful flights (flights that are on time),
We need to calculate the probability that 11 or fewer flights are on time, i.e., $P(X \leq 11)$, where $X$ represents the number of flights that are on time.
Step 2: Use the binomial probability formula.
The probability of exactly $x$ successes in $n$ trials in a binomial distribution is given by the formula:
$$
P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}
$$
However, calculating $P(X \leq 11)$ requires the sum of probabilities from $X = 0$ to $X = 11$. This can be expressed as:
$$
P(X \leq 11) = \sum_{x=0}^{11} P(X = x)
$$
Alternatively, it is easier to calculate the cumulative probability directly using statistical software or a calculator.
Step 3: Calculation using a binomial distribution calculator.
Using a binomial cumulative distribution calculator, we compute the probability $P(X \leq 11)$ for $n = 15$, $p = 0.81$, and $x = 11$.
Step 4: Interpret the result.
The result from the calculator or table will give the cumulative probability, which is approximately:
$$
P(X \leq 11) \approx 0.292
$$
This means that the probability that 11 or fewer of the 15 flights will arrive on time is approximately 29.2%.
Explanation of the approach:
- We are using the binomial distribution because each flight has two possible outcomes: on time (success) or not on time (failure).
- The binomial distribution is appropriate when the trials are independent, and the probability of success is constant across trials.
- The solution involves calculating cumulative probabilities, which can be tedious to do by hand for a binomial distribution. Hence, statistical tools are often used to obtain the result quickly and accurately.
Thus, the probability that 11 or fewer of the 15 selected flights will be on time is about 0.292 or 29.2%.