Question

The computer that controls a bank's automatic teller machine crashes a mean of 0.4 0.4 times per day. What is the probability that, in any seven-day week, the computer will crash less than 4 4 times? Round your answer to four decimal places.

Answer 1

Let X be the number of times per week the computer that controls a bank's automatic teller machine crashes. This can be modelled using Poisson distribution because it satisfies the following properties:

• The experiment results in outcomes that can be classified as successes or failures (crashes or not crashes).
• The average number of successes () that occurs in a specified region (day or week) is known.
• The probability that a success (crash) will occur is proportional to the size of the region (number of days).
• The probability that a success (crash) will occur in an extremely small region is virtually zero.

The computer that controls a bank's automatic teller machine crashes a mean of 0.4 times per day. For 7 days, the mean number of crashes = 7 * 0.4 = 2.8 per week

Thus, X ~ Poisson( = 2.8)

Probability that the computer will crash less than 4 times in a week = P(X < 4)

= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

= 2.8⁰ * exp(-2.8) / 0! + 2.8¹ * exp(-2.8) / 1! + 2.8² * exp(-2.8) / 2! + 2.8³ * exp(-2.8) / 3!

= 0.0608 + 0.1702 + 0.2384 + 0.2225

= 0.6919