A manufacturer of window frames knows from long experience...

A manufacturer of window frames knows from long experience that $5 \%$ of the production will have some type of minor defect that will require an adjustment. What is the probability that in a sample of 20 window frames: a. None will need adjustment? b. At least one will need adjustment? c. More than two will need adjustment?

A manufacturer of window frames knows from long experience that $5 \%$ of the production will have some type of minor defect that will require an adjustment. What is the probability that in a sample of 20 window frames:
a. None will need adjustment?
b. At least one will need adjustment?
c. More than two will need adjustment?

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with a constant probability of success. It is particularly useful when the outcomes of each trial are binary, such as success/failure, and when the trials are independent, meaning the outcome of one does not affect the others.

Bernoulli Trials

A Bernoulli trial is a random experiment with exactly two possible outcomes: success or failure. Each trial in a binomial experiment is a Bernoulli trial, and the probability of success (or failure) remains the same in each trial, which is crucial for applying the binomial distribution.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of it not occurring. This concept is especially useful when calculating the probability of 'at least one' occurrence, as it can be simpler to compute the probability of zero occurrences and subtract it from one.

Cumulative Probability

Cumulative probability is the sum of probabilities of a random variable falling within a specified range. In the context of the binomial distribution, it is used to find the probability of achieving a number of successes up to a certain value, such as determining the likelihood that more than a specific number of successes occur.

Transcript

0:00 All right.

00:01 So we have a manufacturer of window frames.

00:04 Who knows? because they've had so much experience with this, that 5 % of their product will have some minor defect that will require an adjustment.

00:12 And so they sampled 20 window frames.

00:15 And we're going to find some probabilities.

00:17 And is it binomial? is the binomial experiment? there are two outcomes.

00:23 Needs adjustment or not.

00:26 Fixed number of trials.

00:27 Yes, there are 20 and we want some number out of that 20.

00:30 So that's right.

00:31 Probability of success is the same.

00:34 We're assuming it is.

00:35 It's 5%.

00:36 And by the way, success here doesn't mean a good thing.

00:39 It just means success of the 5 % will, the 5 % event will occur.

00:48 They're independent.

00:49 So if you sample one window, it has no effect on the other window, and that's true.

00:53 So it's binomial.

00:55 So we're going to make our little distribution here.

00:58 So combination of n, which is 20, choose x, which is that cell times the probability, which is 0 .05 to the 0 or the x power here in this case, times 1 minus the probability, all raised to the n, which is 20 minus the x power.

01:26 Copy and paste it down...

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