Question

A machine that produces a special type of transistor (a component of computers) has a 10% defect rate. The production is considered a random process where each transistor is independent of the others. What is the probability that the 5th transistor produced is the first with a defect? Which distribution would you use here and why?

Answer 1

Geometric distribution:

Here the variable is X, where X is the number of independent Bernoulli trials needed to get the first success. [each trial with success probability p]

Then the probability that the xth trial is the first success is

The geometric distribution is an appropriate model if the following assumptions are true.

• The phenomenon being modeled is a sequence of independent trials.
• There are only two possible outcomes for each trial, often designated success or failure.
• The probability of success, p, is the same for every trial.

Here, a transistor is checked and identified as "Defect" or "Non-defect". Thus, "Defect"=success and the corresponding success probability is 10%. Thus, p=10/100=0.1.
Also, each transistor is independent of the others. This represnts Bernoulli trials that are independent of each other.

It satisfies all the conditions of geometric distribution. Therefore, we would use geometric distribution.

X ~ Geometric(p=0.1)

Asked is the probability that the 5th transistor produced is the first with a defect i.e. to find P(X=5).
P(X=5)=(1-p)^{5-1} p = (1-0.1)^{5-1} 0.1 = 0.06561

The probability that the 5th transistor produced is the first with a defect is 0.06561.