Given P(A1)=0.60, P(B1 |A1)=0.60, and P(B1 |A2)=0.40, what is the probability of P(A1 |B1) ?

Name: Problem 82, Chapter 3: Statistics for Business and Economics
Uploaded: 2022-01-08T22:09:33-08:00
Duration: 3 min 20 s
Channel: Sheryl Ezze
Description: Given P(A1)=0.60, P(B1 |A1)=0.60, and P(B1 |A2)=0.40, what is the probability of P(A1 |B1) ?

step 1 So we're given the probability of A sub 1 is equal to 0 .6. The probability of the conditional o

Given P(A1)=0.60, P(B1 |A1)=0.60, and P(B1 |A2)=0.40, what...

Key Concepts

Conditional Probability

Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred. It helps us understand how the probability of one event is influenced by the occurrence or non-occurrence of a related event.

Bayes' Theorem

Bayes' Theorem provides a way to update the probability estimate for a hypothesis as more evidence or information becomes available. It connects the conditional probabilities of two events and allows us to invert conditionality — that is, to calculate the probability of a cause given the outcome.

Law of Total Probability

The Law of Total Probability breaks down the probability of an event by considering all possible mutually exclusive scenarios that can lead to that event. This principle is crucial when dealing with conditional probabilities across multiple partitions or cases of a sample space.

Transcript

00:01 So we're given the probability of a sub 1 is equal to 0 .6.

00:06 The probability of the conditional of b1, given that a1 already took place, is 0 .6.

00:12 And the probability of b1 given a sub 2 is equal to 0 .4.

00:19 And we want to find the probability of a1 given b1.

00:26 And so we're going to draw our little chart.

00:30 I don't know what formulas you're given, and i'm just going to start doing it the way i do it.

00:37 And a1 and a2 are going to be complements, and b1 and b2 are going to be complements.

00:43 And so we know that the probability that a happens is this value here, which means the probability of a sub 2 happens is 0 .4, and these two add up to 1.

00:56 Now, the other values we don't know, and these are all intersections here, a1 and b1, a2 and b1, etc.

01:04 So let's use, first of all, this little given information here and fill in some more information.

01:11 So we know that this conditional means the probability of b1 and a1 divided by the probability of a1.

01:22 However, we know that a1, and this, by the way, is equal to 0 .6.

01:28 We know that the probability of a sub 1 is 0 .6, so we can do a substitution.

01:33 And then we can find the probability of that intersection, which is this value in our chart, by taking 0 .6 times 0 .6, and that is 0 .36.

01:42 Therefore, this value will have to be 0 .24.

01:45 These have to add up to 0 .6.

01:47 So now we can go to this information and do a similar analysis.

01:53 We know that this value comes from b1 and a sub 2 divided by the probability of a sub 2...

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