Finite: Probability

Question

A lot contains 120 parts of which 40 are defective and 80 are good. A sample of three different parts is selected at random.

(a) What is the probability that all of the parts selected are good?

(b) What is the probability that at least one of the parts selected is defective?

Answer 1

For this problem, we can consider finding a good part as a success, and finding a defective part as a failure. I am going to assume that you are selecting without replacement.

For part a, you want all three parts to be a success. Thus, you select your first part, and the chance of it being good is 80/120 (or 2/3). Select a second part, and the probability it is good is 79/119, since you have one less good part and one less total part. The third part's probability would be 78/118 (or 39/59).

Since you want each of these things to happen, it's like saying you want the first part to be good AND the second AND the third. Generally, whenever we use the word AND in probability, we multiply.

So, multiplying all three of these probabilities, we get:

2/3 * 79/119 * 39/59 = 6162 / 21063 = 2054 / 7021

For part b, having any parts that are defective would mean ALL the other scenarios besides the ones that you discovered in part a. Thus, you would simply have to subtract what you found in part a from 1.

1 - (2054 / 7021) = 4967 / 7021.

As a note, if the problem mean for you to select with replacement, the process is much easier.

For part a, you would simply do (2/3)^3, which is 8/27. For part b, you would again subtract this from 1, and so you would get 19/27.

Answer 2

The probability that a good part is selected is 2/3.

The probability that a bad part is selected is 1/3.

The probabilities asked for in a) and b) can be determined using the binomial formula:

P(n,k) = C(n,k)p^k * (1-p)^(n-k), where n equals number of trials, k = number of successes (desired outcomes), p = probability of successes and C(n,k) = n!/(k!(n-k)!)

Probability that all parts are selected is good is computed using n=3, k=3, p=2/3

Probability that at least one of the parts selected is defective is 1 - probability that none of the parts are defective, which is equal to 1 - probability that all of the parts are good = 1 - the probability computed for a).