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## Statistics and probability

### Course: Statistics and probability>Unit 7

Lesson 8: Multiplication rule for dependent events

# Dependent probability

Sal finds dependent probabilities like P(A | B) using an example rolling dice. Created by Sal Khan.

## Want to join the conversation?

• The name of the video is Analyzing dependent probability,
Does the math indicate the events are independent?
P(A) = P(A|B)
P(B) = P(B|A)
P(A and B) = P(A) x P(B)
are all true. • @xMcBrennanx:

The fact that they are rolled simultaneously has nothing to do with that. You could have rolled the 6-sided die first, watched a movie, and then rolled the 4-sided die next.

The thing to look for is that the outcome of the 4-sided die has nothing to do with the outcome of the 6-sided die. Whatever the 6-sided die shows, does not make a difference in what the 4-sided die is capable of showing.

I agree with D.A.P. that the title should have been different.
• so all these formulas:
P(A) = P(A|B)
P(B) = P(B|A)
P(A and B) = P(A) x P(B)
only work for independent events am i right in saying that? and they should not work for dependent events, right?! • Let's have an example:
S= {1,2,3,4,5,6,7}
A is a subset of S including even numbers: 2,4,6
B is a subset of S, B= {1,2,3}
P(A)= 3/7
P(B)= 3/7
P(A|B)=1/3 => P(A) is not equal to P(A|B), that means B effects the probability of A. oR you can say: when B happens, the probability of A changes. Therefore, A and B are dependent events.
P(B|A)= 1/3 is also not equal to P(B), because A and B are dependent events. (A effects the probability of B)
P(A and B)= P(A).P(B|A) OR P(A).P(B)?
If I calculate the probability of A first, that means i let A happen first. Remember that A effects the probability of B. So when A happens first, the probability of B must be calculated in the condition of A, so it must be multiplied by P(B|A), not P(B) (the probability of B, doesn't care for A happening or not)
The right formula for P(A and B) when A and B are dependent events is P(A and B)=P(A).P(B|A) = P(B).P(B|A)
In independent events, we don't have to care if A happens first or not happen, because even A happens or not, the probability of B does not change. That's why we have the formula: P(A and B)=P(A).P(B) (A and B are independent)
Hope this helps
• Has anyone else noticed that simplifying each answer as they are completed (within Dependent Probabilities) makes the proceeding problems more confusing? For instance, there's a difference between the answers for questions 1 and 2 simplified and the answers for 3 and 4. Even though they both have the same fractions (1/6 and 1/4).

The trouble I have is using the visual dice chart while solving a problem that requires me to reference the preceding simplified answers. One has to remember that the answer to 1 is always 4/24 (The whole visual dice chart) even if you simplify it to 1/6. While answer 3 is always 1/6 because it is only dealing with the bottom most horizontal line on the visual dice chart. I think if the answers are kept unsimplified until the very end, it will make the visual dice chart clearer, and the previous questions more helpful as a reference. • Why isn't the uniform distribution mentioned at all? Its very tedious having to count tiles out repeatedly for the same question over and over, when simple dice probabilities can be less tediously done with a simple formula.
(1 vote) • What is the conceptual reasoning behind multiplying the simple probabilities with the dependent probabilities I.E.: P(A) x P(A | B) ? I'm having trouble rationing out the reason behind the equations when the problem is independent. What would it do when it's dependent?
(1 vote) • Probably a better way to understand it is going the other way. The definition of conditional probability is:

P(A|B) = P( A ∩ B) / P(B)

In this, we are scaling the intersection by the probability of B. Think of a Venn Diagram with two circles for events A and B. Then, when we add the condition on B, we are saying that we know B already happened. So we want to consider ONLY the area that is inside of B. From that perspective, the probability of A, given that B occurred, is just the proportion of the B circle which also contains A.

This is nothing more than a ratio, the same thing you'd do if you were to calculate, say, the percentage of girls in a Stat class: We'd count how many girls are in the class, and divide that by how many people are in the class. So in the case of conditional probability, we'd find the probability of the intersection, and divide by the probability of the full circle of B, to get that formula up above.

The idea of multiplying the conditional probability by the probability of the condition is simply doing this process in reverse. Look back up at that formula. The denominator, P(B), is just a probability, just a number. We can rearrange that formula by, say, multiplying both sides of the equation by P(B). When we do that, we get:

P(A|B) x P(B) = P( A ∩ B)

Or put another way: We obtained the conditional probability in the first place by scale the probability of the intersection by the probability of B. To get the probability of the intersection, all we have to do us "upscale" the conditional probability.
• Is it a coincidence that in the example, P(A) = P(A | B) and P(B) = P(B | A)? • As to or so, if she's rolling simultaneously, it seems to me that the question "what is the probability that she rolls doubles given that the 4 sided die is 4?" just means "What is the probability that she rolls a 4 and a 4?", which is 1/24, being 1 outcome out of 24.

Same for all the following questions.

Ie, it's one outcome out of 24 possible ones.

If she was rolling the dice one after the other, THEN given that she rolled a 4 first (a probability of 1/4, which is no longer relevant because it is a certainty), then the probability of getting a 4 on the 6 sided die is the only question, and THAT is 1/6. • why my brain thinks that rolling these two dies are independent events? • Quick question, at when Sal is talking about P(A | B), the calculation would be
P (A and B) = P(A) * P(A | B) or
P (1/6 * 1/4) = 1/6 * P(A | B) or
P (1/24) = 1/6 * P(A | B) and then dividing both sides by 1/6
You get: 1/4 = P(A | B) or the answer is 1/4, but the real answer is 1/6... Am I doing the formula wrong? Why is this not working?  