Question

1 (a) How many samples are possible consisting of 5 bolts from a panel with 9 bolts? (Generally, the bolts are picked for a sample with repetition of bolts not allowed. Also, the order in which the bolts are picked does not matter).

(b) Now, assume a panel with 9 bolts contains 2 bad bolts. How many samples of 5 of the bolts contain only good bolts? (Consider a bolt to be good if it is not bad).

(c) A turnpike authority's quality control procedure consists of accepting a panel if all of the 5 inspected bolts from a panel with 9 bolts are good. Find the probability that such a panel containing 2 bad bolts is accepted. (In this question, please report your answer as a decimal accurate to three significant figures, such as .729, and not as a percentage, such as 72.9%).

(d) A turnpike authority's quality control procedure consists of rejecting a panel if one or more of the 5 inspected bolts from a panel with 9 bolts is bad. Find the probability that such a panel containing 2 bad bolts is rejected. (In this question, please report your answer as a decimal accurate to three significant figures, such as 0.729, and not as a percentage, such as 72.9%).

(e) If the turnpike authority uses a quality control procedure that has a 15% probability of accepting a panel with 2 bad bolts to inspect 300 panels, with 2 bad bolts in each panel, what is the expected (or mean) number of panels that will be accepted?

Answer 1

Combination function nCr =

1 This is a combination question since the bolts are picked for a sample with repetition of bolts being not allowed. Also, the order in which the bolts are picked does not matter.

We have 9 bolts and 5 have to be selected. Ways of doing that are

9C5 = (using the formula above)

Final Ans: 126 ways

(b) If we split the 9 bolts into good and bad then we will have 2 bad bolts and 7 good.

If we want all the 5 selected to be good bolts then we will have 7 options to choose from.

No. of ways to choose only good = 7C5

Final Ans: 21

(c) For the authority to accept the panel we will need the sample to have all 5 bolts good. No. of ways of doing that is calculated above. This is our desired outcome.

Possible ways of selecting 5 out of 9 bolts is in Q1. This is our total possible outcomes.

Therefore probability that panel will be accepted =

=

Final Ans: 0.167

(d) For the authority to reject the panel sample has to have at least 1 bad bolt. Since there are only 2 bad bolts, the panel can have 1 or 2 bad bolts. For a bad bolt to be selected it has to be chosen from the 2 bad bolts and the remaining will be chosen from the 7 good bolts.

ways to select 1 bad bolt and 4 good = 2C1 * 7C4 = 70

ways to select 2 bad bolts and 3 good = 2C2 * 7C3 = 35

These will be added since we don't know whether 1 or 2 bad bolts will be chosen

Therefore probability that panel will be rejected =

=

Final Ans: 0.833

(e) If the turnpike authority uses a quality control procedure that has a 15% probability of accepting a panel with 2 bad bolts to inspect 300 panels, with 2 bad bolts in each panel, what is the expected (or mean) number of panels that will be accepted?

Ans: For calculating the expected value we simple multiply the probability with the number of trials(panels).

Expected Number of panels to be expected = 300 * 0.15

Final Ans: 45