Question

a computer technician notes that 40% of computers fail because of the hard drive. if he repairs many computers a day, what is the probability that the first computer that has failed due to the hard drive is his 4th computer of the day? 0.0384 0.0864 0.1296 0.3456

Explanation:

Step1: Identify the probability of non - hard drive failure

The probability that a computer does not fail due to the hard drive is \(p'=1 - 0.4=0.6\).

Step2: Calculate the probability for the first three non - hard drive failures and the fourth hard drive failure

We use the geometric probability formula \(P(X = k)=(1 - p)^{k - 1}p\), where \(p = 0.4\) (probability of hard drive failure) and \(k = 4\) (the number of trials).
Substitute the values into the formula: \(P(X = 4)=(0.6)^{3}\times0.4\).
First, calculate \((0.6)^{3}=0.6\times0.6\times0.6 = 0.216\).
Then, \(0.216\times0.4=0.0864\).

Answer:

\(0.0864\) (the second option)