There is a [198/pi integral^infinity_0 1/1 + x^2 dx]% chance that the following

Question

There is a [198/pi integral^infinity_0 1/1 + x^2 dx]% chance that the following problem will appear on the final exam. A commonly used practice of airline companies is to sell more tickets than actual scats to a particular flight because customers who buy tickets do not always show up for the flight. Suppose that an airline sells 200 tickets for a certain flight, which has only 190 seats. If on the average 4 % of purchasers of airline tickets do not appear for the departure of their flight, find the a. probability that everyone who shows up for this flight will have a seat. b. expected number of passengers with no seat available for this flight. Assume that passengers miss their flight independently of other passengers on the same flight (which is not true in general when families are involved).

Explanation / Answer

Solution:-

a) The probability that everyone who shows up for flight will have a seat is 0.2808.

Probability that passenger will show up for flight = 0.96

n = 200, x = 190

By applying binomial distribution:-

P(x, n, p) = nCx*p x *(1 - p)(n - x)

P(x < 190) = 0.2808

b) Expected number of passengers with no seat available for this flight is 2.

Expected number of passenger arriving for flight = n × p

E(x) = 200 × 0.96

E(x) = 192

There are only 190 seats so 2 passengers will not have seat available in this flight.

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