Answer:
a) Check Explanation
b) Probability that 11 out of the 17 randomly selected flights are on time = P(X = 11) = 0.0680
c) Probability that fewer than 11 out of the 17 randomly selected flights are on time
= P(X < 11) = 0.0377
d) Probability that at least 11 out of the 17 randomly selected flights are on time
= P(X ≥ 11) = 0.9623
e) Probability that between 9 and 11 flights, inclusive, out of the randomly selected 17 are on time = P(9 ≤ X ≤ 11) = 0.1031
Step-by-step explanation:
a) How to know a binomial experiment
1) A binomial experiment is one in which the probability of success doesn't change with every run or number of trials. (Probability of each flight being on time is 80%)
2) It usually consists of a number of runs/trials with only two possible outcomes, a success or a failure. (It's either the flights are on time or not).
3) The outcome of each trial/run of a binomial experiment is independent of one another.
All true for this experiment.
b) Probability that exactly 11 flights are on time.
Let X be the random variable that represents the number of flights that are on time out of the randomly selected 17.
Binomial distribution function is represented by
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
n = total number of sample spaces = 17 randomly selected flights
x = Number of successes required = number of flights required to be on time
p = probability of success = Probability of a flight being on time = 80% = 0.80
q = probability of failure = Probability of a flight NOT being on time = 1 - p = 1 - 0.80 = 0.20
P(X = 11) = ¹⁷C₁₁ (0.80)¹¹ (0.20)¹⁷⁻¹¹ = 0.06803777953 = 0.0680
c) Probability that fewer than 11 flights are on time
This is also computed using binomial formula
It is the probability that the number of flights on time are less than 11
P(X < 11) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.0376634429 = 0.0377
d) Probability that at least 11 out of the 17 randomly selected flights are on time
This is the probability of the number of flights on time being 11 or more.
P(X ≥ 11) = P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17)
= 1 - P(X < 11)
= 1 - 0.0376634429
= 0.9623365571 = 0.9623
e) Probability that between 9 and 11 flights, inclusive, are on time = P(9 ≤ X ≤ 11)
This is the probability that exactly 9, 10 or 11 flights are on time.
P(9 ≤ X ≤ 11) = P(X = 9) + P(X = 10) + P(X = 11)
= 0.0083528524 + 0.02672912767 + 0.06803777953
= 0.1031197592 = 0.1031
Hope this Helps!!!
