Question

A. According to an airline, flights on a certain route are NOT on time 15% of the time. Suppose 10 flights are randomly selected and the number of NOT on time flights is recorded. Find the probability of the following question. At least 3 flights are not on time.

B. According to an airline, flights on a certain route are NOT on time 15% of the time. Suppose 10 flights are randomly selected and the number of NOT on time flights is recorded. Find the probability of the following question. At the most 8 flights are on time.

c. According to an airline, flights on a certain route are NOT on time 15% of the time. Suppose 10 flights are randomly selected and the number of NOT on time flights is recorded. Find the probability of the following question. In between 6 and 9 flights are on time.

Answer 1

a)

Here, n = 10, p = 0.15, (1 - p) = 0.85 and x = 3
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 3).
P(X >= 3) = (10C3 * 0.15^3 * 0.85^7) + (10C4 * 0.15^4 * 0.85^6) + (10C5 * 0.15^5 * 0.85^5) + (10C6 * 0.15^6 * 0.85^4) + (10C7 * 0.15^7 * 0.85^3) + (10C8 * 0.15^8 * 0.85^2) + (10C9 * 0.15^9 * 0.85^1) + (10C10 * 0.15^10 * 0.85^0)
P(X >= 3) = 0.1298 + 0.0401 + 0.0085 + 0.0012 + 0.0001 + 0 + 0 + 0
P(X >= 3) = 0.1797

b)
Here, n = 10, p = 0.15, (1 - p) = 0.85 and x = 8
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <= 8).
P(X <= 8) = (10C0 * 0.15^0 * 0.85^10) + (10C1 * 0.15^1 * 0.85^9) + (10C2 * 0.15^2 * 0.85^8) + (10C3 * 0.15^3 * 0.85^7) + (10C4 * 0.15^4 * 0.85^6) + (10C5 * 0.15^5 * 0.85^5) + (10C6 * 0.15^6 * 0.85^4) + (10C7 * 0.15^7 * 0.85^3) + (10C8 * 0.15^8 * 0.85^2)
P(X <= 8) = 0.1969 + 0.3474 + 0.2759 + 0.1298 + 0.0401 + 0.0085 + 0.0012 + 0.0001 + 0
P(X <= 8) = 0.9999

c)

Here, n = 10, p = 0.15, (1 - p) = 0.85, x1 = 6 and x2 = 9.
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(6 <= X <= 9)
P(6 <= X <= 9) = (10C6 * 0.15^6 * 0.85^4) + (10C7 * 0.15^7 * 0.85^3) + (10C8 * 0.15^8 * 0.85^2) + (10C9 * 0.15^9 * 0.85^1)
P(6 <= X <= 9) = 0.0012 + 0.0001 + 0 + 0
P(6 <= X <= 9) = 0.0013