Statistics Problem

Question

A hotel in South Beach has 30 rooms to rent. The probability for each room to be rented for the next weekend is 0.90. What is the prob that exactly 23 rooms will be rented? ANSWER = .018

ANother hotel has 500 rooms to rent. use the same constant probability of 0.90. what is the approximate probability that at least 455 rooms will be rented in this hotel?

Answer 1

Use binominal formula:

n!

---- p^x (1-p)^n-x

(n-x)!

where n is sample size, p is probability of success and x is number of successes

here success is the room being rented.

Applying binomial formula to this problem P(x = 23):

30! (0.9)^23 (0.1)^7

-----

23! 7!

Use normal approximation to binomial formula:

np = 500 * 0.9 = 469.8

sqrt(npq) = sqrt(500 * 0.9 * 0.1) = 6.708

where q = 1 - p

z = (x - np)/sqrt(npq) is distributed N(0,1)

P( x > 455) =

P (z > (455 - 460.8)/6.708) = P(z > -0.86) = 1 - P(z < -0.86( = 1 - 0.1949 = 0.8051