Final answer:
To find the probability that the inspector has tested 7 times in total, we can use the geometric distribution. The formula for geometric distribution is P(X = k) = (1 - p)^(k-1) * p, where X is the number of trials needed to find the first success, p is the probability of success, and k is the desired number of trials. By substituting the given values and solving the equation, we find that the probability is 0.021.
Step-by-step explanation:
To solve this problem, we can use the concept of geometric distribution. The geometric distribution is used to model the number of trials needed to get the first success in a sequence of independent Bernoulli trials. In this case, the success is finding a defective product and the trials are the number of random picks.
Let's define X as the number of trials needed to find the first defective product. We want to find P(X = 7), which is the probability that the inspector has tested 7 times in total.
Using the formula for geometric distribution, we have P(X = 7) = (1 - p)^(7-1) * p, where p is the probability of picking a defective product.
Given that there are 7 acceptable products and 3 defective products out of 10 transistors, the probability of picking a defective product is p = 3/10.
Substituting p = 3/10 into the formula, we can calculate P(X = 7) as (1 - 3/10)^(7-1) * (3/10) = 0.021 (rounded to three decimal places).