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# General multiplication rule example: independent events

We can use the general multiplication rule to find the probability that two events both occur when the events are independent. Created by Sal Khan.

## Want to join the conversation?

• I am curious why we cannot answer the question like the following.

The probability of getting silk on the first spin is 1/6.
The probability of getting silk on the second spin is 1/6.
So the probability that Doug or Maya will get silk is 2/6, or 1/3.

So then the probability of neither of them getting silk must be the inverse of this or 2/3, or 0.66666666666?

However the probability obtained by simply multiplying 5/6(5/6) is 25/36 or 0.6944444444444. Almost the same but not exactly.

So, it seems that we can't simply inverse the probability of the positive event to get the negative event? Why not?
• Interesting question!

“Inversing” the “positive” event to get the “negative” event is not the mistake. Instead, the mistake is assuming that the probability of the “positive” event of either Doug or Maya getting silk is 1/6 + 1/6. This is incorrect because these two events of silk for Doug and silk for Maya could both occur (they are not mutually exclusive). So this calculation counts twice, instead of once, the probability that both events occur. So we must correct this by subtracting the probability that both occur, which is 1/6 * 1/6 = 1/36.

So the probability of the “positive” event is 1/6 + 1/6 - 1/36 = 11/36. Then the probability of the “negative” event is indeed 1 - 11/36 = 25/36, which matches 5/6 * 5/6.

Have a blessed, wonderful Christmas!
• Probability that Maya and Doug both get silk should be 1/6 x 1/6 = 1/36.
The probability that they both don't get silk is 25/36.
Shouldn't probability that they both get silk and they both don't get silk be 1 but 1/36 + 25/36 is not equal to 1. Please clarify.
(1 vote)
• They do not add up to 1 because there are other possibilities you have not taken into account. Maya could get silk and Doug does not, which is 1/6 x 5/6 = 5/36. Doug could get silk and Maya does not, which is also 1/6 x 5/6 = 5/36.

probability that neither gets silk + probability that only Maya gets silk + probability that only Doug gets silk + probability that both get silk
= 25/36 + 5/36 + 5/36 + 1/36
= 36/36
= 1
• Why is the second draw not 4/5 assuming silk was not drawn in the first draw? That is, why is one of the options not taken off the wheel after the first spin? The question isn’t very clear that both could draw the same material.
• The problem did not indicate that the option can be taken off the wheel after Maya's turn. If that was the case then Doug's turn would be considered as dependent to the 1st event (ie. Maya's turn) as the wheel has been altered. For the purpose of this problem, it is important that the wheel is fair and unaltered in the sequence of events.
• I'm curious won't Doug have total options of materials = 5, since Maya already chose one(Not silk). So won't it be 5/6 multiply by 4/5, which results in 2/3?
• There is no rule that they must have different materials. They could very well both get wood, or both get plastic, and so on.
• why the condition P(A).P(B/A), is just equal of P(B) is it because the events are dependant or independent?
• If P(B|A) and P(B) are equal, then the events are independent. This is because the outcome of A does not affect the probability of B.
• What happens if the probability is both theoretical and experimental?
• They are mutually exclusive, two different things. While there is a possibility that they are equal, they are two different things. Theoretical is what should happen, experimental is what happens when you run an experiment.
• what does it mean by generation of multiplication
• If 𝐴 and 𝐵 are independent events, then 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)⋅𝑃(𝐵)

The general multiplication rule works for both independent and dependent events:
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)⋅𝑃(𝐵 | 𝐴)
• If this were dependent so that when a person got an outcome, it could not be used again, how would the expression be written?