Question

The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best rate with 80 % of its flights arriving on time. A test is conducted by randomly selecting 15

Southwest flights and observing whether they arrive on time. (a) Find the probability that exactly 7 flights arrive late.

Answer 1

Explanation:

Assume that the event of a flight arriving on time/late is independent of each other.

P(7 flights arrive late) = 15C8 * (0.8)⁸(0.2)⁷ ≈ 0.0138.

Answer 2

P(x = 7) = 0.0138 (4 d.p.)

Explanation:

To find the probability that exactly 7 out of 15 Southwest flights arrive late when the on-time rate is 80%, we can use the binomial probability formula:

`\boxed{\displaystyle P(x = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^(n - k)}`

where:

• n is the number of trials.

• k is the number of successful trials.

• p is the probability of success.

In this case:

• The number of trials is the number of flights, so n = 15.

• The successful trials are the number of flights arriving late, so k = 7.

• The probability of "success" is the probability of a flight arriving late, so p = 1 - 0.80 = 0.2.

Calculate the probability by substitute the values into the formula:

`\displaystyle P(x = 7) = \binom{15}{7} \cdot 0.2^7 \cdot (1 - 0.2)^(15 - 7)`

`\displaystyle P(x = 7) = \binom{15}{7} \cdot 0.2^7 \cdot (0.8)^(8)`

`P(x = 7) = (15!)/(7!(15-7)!)\cdot 0.2^7 \cdot (0.8)^(8)`

`P(x = 7) = 6435\cdot 0.0000128 \cdot 0.16777216`

`P(x = 7) = 0.082368 \cdot 0.16777216`

`P(x = 7) = 0.01381905727488`

`P(x = 7) = 0.0138\; \sf (4\;d.p.)`

Therefore, the probability that exactly 7 out of 15 Southwest flights arrive late is approximately 0.0138 (rounded to four decimal places).