Human visual inspection of solder joints on printed circuit...

Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad nonwetting, knee visibility, voids) and even the degree to which a joint possesses one or more of these defects. Consequently, even highly trained inspectors can disagree on the disposition of a particular joint. In one batch of $10,000$ joints, inspector A found 724 that were judged defective, inspector B found 751 such joints, and 1159 of the joints were judged defective by at least one of the inspectors. Suppose that one of the $10,000$ joints is randomly selected. (a) What is the probability that the selected joint was judged to be defective by neither of the two inspectors? (b) What is the probability that the selected joint was judged to be defective by inspector B but not by inspector A?

Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad nonwetting, knee visibility, voids) and even the degree to which a joint possesses one or more of these defects. Consequently, even highly trained inspectors can disagree on the disposition of a particular joint. In one batch of $10,000$ joints, inspector A found 724 that were judged defective, inspector B found 751 such joints, and 1159 of the joints were judged defective by at least one of the inspectors. Suppose that one of the $10,000$ joints is randomly selected.
(a) What is the probability that the selected joint was judged to be defective by neither of the two inspectors?
(b) What is the probability that the selected joint was judged to be defective by inspector B but not by inspector A?

Key Concepts

Complement Rule

The complement rule states that the probability of an event not occurring is equal to one minus the probability that the event does occur. This rule is especially useful for finding the probability that none of several events occur, by considering the complement of the event that at least one occurs.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a counting method used to calculate the probability or number of elements in the union of multiple sets by adding the sizes of each set and subtracting the sizes of all pairwise (and higher order) intersections. This principle is crucial for correctly handling overlapping events when determining the total probability of combined occurrences.

Set Operations in Probability

Set operations, including union, intersection, and complement, are used to represent and analyze events in probability. They allow us to combine or exclude events and help in formulating problems where outcomes may overlap, enabling the precise calculation of probabilities in complex scenarios.

Probability

Probability is the measure of the likelihood that an event will occur, calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. It provides the fundamental basis for assessing risk, reliability, and performance in various contexts by quantifying uncertainty.

Transcript

00:01 Okay, so to start off with this problem, we are first going to define a few parameters.

00:08 We're going to have n be the total number of solder joints, so that will be 10 ,000.

00:18 We have n, a, represent the number of joints judged effective by inspector a.

00:27 So that's 724.

00:32 We're going to have nb.

00:38 Be the number of joint judge defective by inspector b, 751.

00:46 And then we're going to have the number of joints judged effective.

00:53 Oh, sorry, that be pretty bad.

00:56 Judge defective by at least one of the inspectors, and that's 1 ,159.

01:04 So if you notice with this, 7509 ,000.

01:07 751 and 724 don't add up to 1 ,159.

01:13 That's because some of these joints, they both agree that they're defective.

01:19 Some they have differing opinions.

01:22 So this number right here only means that at least one of them deems the joint defective, but it can be that they both agree.

01:32 Okay.

01:33 So let me adjust this a little bit.

01:38 Part a, what is the probability that the selected joint was judged effective by neither of the inspectors? okay, so this is actually pretty straightforward.

01:50 So to get the probability of that, we would need to take the total number of joints minus the number deemed defective by at least one of the inspectors, and then divide all that by the total again.

02:21 So that will give us 10 ,000, excuse me, 10 ,000 minus 1 ,159 over 10 ,000.

02:40 And then so if you do all that math, it will get out to 0 .841 or expressed as a percentage.

02:57 It would be 80.

03:00 Oh, sorry, that 8 was terrible.

03:02 88 .41%.

03:08 Okay.

03:10 And then part b, we have what is the probability that the selected joint was judged to be defective by inspector b, but not by inspector a...

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