The employees parking lot at the Buss Stout Fence Company...

The employees parking lot at the Buss Stout Fence Company has 50 percent U.S. cars, of which 15 percent are compact; 30 percent of the cars are European, of which 40 percent are compact; and 20 percent of the cars are Japanese, of which 60 percent are compact. If a car is randomly selected from the lot, calculate (a) The probability it is a compact. (b) Given that the car is a compact, the probability that it is European.

The employees parking lot at the Buss Stout Fence Company has 50 percent U.S. cars, of which 15 percent are compact; 30 percent of the cars are European, of which 40 percent are compact; and 20 percent of the cars are Japanese, of which 60 percent are compact. If a car is randomly selected from the lot, calculate
(a) The probability it is a compact.
(b) Given that the car is a compact, the probability that it is European.

Key Concepts

Total Probability Rule

The Total Probability Rule is used to compute the probability of an event by considering all the different mutually exclusive ways in which that event can occur. This involves partitioning the sample space into distinct components, calculating the probability of the event within each component, and then summing these probabilities to obtain the overall probability.

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event has occurred. It is an essential concept in probability theory because it helps in understanding how the occurrence of one event influences the likelihood of another event.

Bayes' Theorem

Bayes' Theorem provides a method to update the probability estimate for an event based on new evidence or information. It relates the conditional and marginal probabilities of events and is particularly useful when one needs to reverse conditional probabilities or determine the probability of a cause given an observed effect.

Transcript

00:01 So we know that there's approximately 5 % of people take the public transit to work.

00:07 And they said they're going to give a discount to at least 30 people at these companies.

00:13 And we have three conditions.

00:16 We have that the company has 250 people, the company has 500 people, or a company has 1 ,000 people.

00:28 And so we want to see what's the likelihood of getting.

00:32 Greater than are equal to 30 in each of these settings.

00:45 And first of all, we can see that n times p, if we find the mean of each of these, the 250 times .05, the mean number of people is 12 .5 here, and the standard deviation is times .95, and then to the .5, comes out to be 3 .446.

01:13 And so converting this value to, and let's just draw a line down here, creating this value to a z value, and we can quick do the continuity correction.

01:24 We'd have 29 .5 minus that mean divided by that standard deviation.

01:30 Let's see what that ends up being.

01:33 And click store that value.

01:35 So we'd have that 29 .5 minus 30.

01:39 Whoops, minus the 12 .5, divided by that standard deviation.

01:48 And that z value is like 4 .9.

01:52 So z is greater than or equal to 4 .93.

01:55 And this probability is going to be very close to zero.

01:58 So it's very little chance that that's going to happen.

02:01 Now, there is a probability there, but it's just very close to zero.

02:04 So that's not very likely.

02:06 Now, what if we have 500 employees? well, we know this distribution will also be approximately normal.

02:11 And if we take 5 % of 500, the mean number wouldn't be expected to be 25, and the standard deviation would be times 0 .95, and then square root that.

02:26 The standard deviation there has reduced down, or has changed to, gotten bigger than 4 .873.

02:35 And now we'll find that z value.

02:40 Again, i'll use the continuity correction just to be a little or accurate...

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