Math Problem Statement

From experience, an airline knows that only 70% of the passengers booked for a certain flight actually show up. If passengers are randomly selected, find the probability that at most 5 of them show up. Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

Solution

We can model this situation using the binomial distribution. The number of passengers showing up follows a binomial distribution because each passenger either shows up or does not (two possible outcomes), and each passenger is independent of the others.

The binomial distribution is defined as:

P(X=k)=(nk)pk(1p)nkP(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}

where:

  • nn is the number of passengers booked.
  • pp is the probability that a passenger shows up (in this case, 0.70).
  • kk is the number of passengers who show up.
  • (nk)\binom{n}{k} is the binomial coefficient: n!k!(nk)!\frac{n!}{k!(n-k)!}.

We are asked to find the probability that at most 5 passengers show up, which means calculating:

P(X5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)P(X \leq 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

Steps:

  1. Identify the parameters:
    • The probability of a passenger showing up: p=0.70p = 0.70
    • The probability of a passenger not showing up: 1p=0.301 - p = 0.30
    • The number of trials (passengers booked): nn (but this value is not provided directly, so if it's missing we cannot continue).

Could you please specify the number of passengers booked? That way, I can proceed with the calculations.

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Math Problem Analysis

Mathematical Concepts

Binomial Distribution
Probability

Formulas

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

Theorems

Binomial Theorem

Suitable Grade Level

Grades 11-12 or College Level