Computer keyboard failures can be attributed to electrical...

Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects. (a) How many ways are there to randomly select 5 of these keyboards for a thorough inspection (without regard to order)? (b) In how many ways can a sample of 5 keyboards be selected so that exactly 2 have an electrical defect? (c) If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?

Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects.
(a) How many ways are there to randomly select 5 of these keyboards for a thorough inspection (without regard to order)?
(b) In how many ways can a sample of 5 keyboards be selected so that exactly 2 have an electrical defect?
(c) If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?

Key Concepts

Combinations

Combinations refer to selections of items where the order of selection is irrelevant. The formula for combinations, typically represented by the binomial coefficient, helps calculate the number of ways to choose a subset from a larger set.

Counting Techniques

Counting techniques involve methods to determine the number of ways events or selections can occur. These methods include both arrangements where order matters, such as permutations, and arrangements where order does not matter, such as combinations.

Binomial Coefficient

The binomial coefficient, often read as 'n choose k', computes the number of ways to select k items from a collection of n distinct items without considering order. It is defined mathematically as n! divided by the product of k! and (n?k)!.

Probability

Probability quantifies the likelihood of an event occurring and is calculated as the ratio of favorable outcomes to the total number of possible outcomes. In combinatorial contexts, it often involves dividing the number of desirable combinations by the total number of combinations.

Hypergeometric Distribution

The hypergeometric distribution is used in scenarios involving sampling without replacement from a finite population that contains two distinct types of items. It is particularly useful for determining probabilities of a specific number of successes in the sample drawn from the population.

Transcript

00:01 Okay, so we have 25 keyboards that are busted, six have electrical defects, and 19 have mechanical, and we're asking ourselves, how many ways can we choose five? and they tell us order doesn't matter for this question.

00:18 So if order doesn't matter, then we get to choose, well, combinations.

00:24 So we have 25 keyboards.

00:28 Let me make that a little bit smaller.

00:31 So we have 25 keyboards, and we're going to say choose five of them.

00:40 So we're going to use that program in our calculator that will do 25, choose five.

00:45 And this is going to be equal to 5 -1, my mistake, 5 -3 -1 -3 -0.

00:58 So there are 53 ,000 possibilities if you have 25 keyboard that you just select five at random.

01:08 That's quite a few.

01:12 So for b, they ask, we want to know how many different ways that you have exactly two have an electrical defect.

01:21 So that means three are mechanical and two are electrical.

01:37 So we have to pick intentionally now three mechanical problems.

01:43 So there are 19 ones which have mechanical defects.

01:47 So we're going to say 19 choose three.

01:52 We're going to do that on our calculator, 19, math, probability, probability down to ncr.

02:05 And 19 choose three will give us 969.

02:15 So we have 969 possibilities for if we have 19 keyboards to select three of them.

02:24 Now we have to pick two electrical ones, and we have six that have electrical defects.

02:30 So we're going to have six choose two.

02:38 So again, we're going to use our calculator to do this for us.

02:40 You could do it out by hand.

02:41 It depends on what is required of you.

02:46 But that should give us a 15 using a calculator.

02:54 Now we have to deal with these two numbers.

02:58 So if we have these nearly 1 ,000 mechanical issues and just 15 electrical possible combinations, so it ends up being that we're going to multiply these numbers together because we can have a whole bunch of different combinations therein.

03:15 So we're going to have 9, 6, 9 times 15...

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