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gtnhenbr [62]
3 years ago
13

Suppose that an airline overbooks seats on their flights. In particular, it sells 300 tickets for a flight when there are only 2

70 seats available. On average, we expect 15% of those with tickets to not show up. What is the probability that we will have enough seats for everyone who shows up
Mathematics
1 answer:
vladimir1956 [14]3 years ago
8 0

Using the <u>normal approximation to the binomial</u>, it is found that there is a 0.994 = 99.4% probability that we will have enough seats for everyone who shows up.

In a normal distribution with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

  • It measures how many standard deviations the measure is from the mean.  
  • After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
  • The binomial distribution is the probability of <u>x successes on n trials</u>, with <u>p probability</u> of a success on each trial. It can be approximated to the normal distribution with \mu = np, \sigma = \sqrt{np(1-p)}.

In this problem:

  • 15% do not show up, so 100 - 15 = 85% show up, which means that p = 0.85.
  • 300 tickets are sold, hence n = 300.

The mean and the standard deviation are given by:

\mu = np = 300(0.85) = 255

\sigma = \sqrt{np(1-p)} = \sqrt{300(0.85)(0.15)} = 6.185

The probability that we will have enough seats for everyone who shows up is the probability of at most <u>270 people showing up</u>, which, using continuity correction, is P(X \leq 270 + 0.5) = P(X \leq 270.5), which is the <u>p-value of Z when X = 270.5</u>.

Z = \frac{X - \mu}{\sigma}

Z = \frac{270.5 - 255}{6.185}

Z = 2.51

Z = 2.51 has a p-value of 0.994.

0.994 = 99.4% probability that we will have enough seats for everyone who shows up.

A similar problem is given at brainly.com/question/24261244

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