Question

Question 5: (10 points) A stock brokerage has four computers that are used for making trades on the New York Stock Exchange. The probability that a computer fails on any single day is 0.005. Failures occur independently. Any failed computers are repaired after the exchange closes, so each day can be considered an independent trial. (a) What is the probability that all four computers fail on one day? (3 points) (b) What is the probability that at least one computer fails on a day? (7 points)

Answer 1

Answer:-

Given That:-

A stock brokerage has four computers.

The probability that a computer fails on any single day is 0.005.

Failures occur independently.

Let X denotes number of computers fail on a day.

Here, $X \sim Binom(n = 4, p = 0.005)$

The PMF of X can be written as,

$P(X = x) = \binom{n}{x} p^{x} (1 - p)^{n - x}$

=  $\binom{4}{x}(0.005)^{x} (0.995)^{4 - x}, x = 0, 1, 2, 3, 4$

(a) Compute the probability that all four computers fail on one day.

$P(X = 4) = \binom{4}{4}(0.005)^{4} (0.995)^{4 - 4}$

=

(1{0.005)+(1)

= $(0.005)^{4}$

= $6.25 \times 10^{-10}$

  $\approx 0.0000$

Therefore, probability that all four computers fail on one day is approximately, 0.0000.

(b) Compute the probability that at least one computer fails on a day.

$P(X \geq 1) = 1 - P(X < 1)$

=

= $1 - \binom{4}{0} (0.005)^{0} (0.995)^{4 - 0}$

=

=

$\approx 0.0199$

Therefore, probability that at least one computer fails on a day is 0.0199.