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# Interpreting general multiplication rule

We can express the probability that two events both occur symbolically using the general multiplication rule, and we can interpret probability statements that are expressed symbolically. Created by Sal Khan.

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• Wait, but didn't Sal say that whatever the contestants landed on wouldn't be taken out, and so each contestant has an equally likely possibility of landing on kale? Or is the scenario just different for the seond question?
• Oof, I put the answer in the comments, but I'll type it again. Yes, it is true that he said that. But the question asked only the interpretation of the said statement. It doesn't necessarily mean that the statement is true. If it helps to think about it as the scenario is different, you can!

Hope this helps!
• In previous videos, Sal uses the equation P(A ∩ B) = P(A)P(B | A). In this one, he uses P(A ∩ B) = P(A | B)(B). Are both these equations equivaent?

If they are, then I'm confused on the quesitons in the upcoming exercise. "P(B1 ∩ B2) = P(B1)P(B2 | B1). What does P(B1 ∩ B2) represent?" If both aforementioned equationsare equivalent, then surely the options:
The probability that the first spin lands on bankrupt given the second spin lands on bankrupt
and
The probability that the second spin lands on bankrupt given the first spin lands on bankrupt
are equivalent, but apparently only the latter is correct.

If they aren't, then what makes Sal choose which equation to choose? I don't recall him going over when to use which.
• When evaluating the result, it's equivalent. P(A ∩ B) = P(A)P(B | A) = P(A | B)(B)

However if you consider the meaning behind the equation, it's different. It's like 5 * 6 means adding 5s 6 times, and 6 * 5 means adding 6s 5 times. You will get 30 either way, but they represent different things.

It depends on how you write which one first in the probability, but one should be P(A ∩ B) and other should be P(B ∩ A).

For the exercise, you should consider the conditional probability. P(B2 | B1) means given B1 and the probability of B2 occurring, which will be the latter.
• I need some help with the 2nd question:
Since the 2 events are independent, is the 'given' necessary in the right part of the equal sign?

In other words, P(K2 | K1^C) is the same as P(K2), right?

Thank you!
• The 2 events are not independent, if the first contestant took spinach, then the second contestant will have no possibility to take spinach wich means the second contestant's possibillity to land on kale changed from 1/6 to 1/5. Therefore, P(K2|K1^C) and P(K2) are definitely not the same!
Like the bag of marbles example, the marble was removed from selection space once picked and the remaining total quantity was x-1.

In this example the option of kale wasn't removed on the first selection K_1^C = "first contestant does not land on kale".
The equation was:

P(K_1^C and K_2) = P(K_1^C) x P(K_2 | K_1^C)

It wasn't mentioned in the question that the resources were limited to one serving. Even if there was only one portion of each product, the option is still there on the spinning wheel. As far as I understand, that means that the probability of landing on any option remains 1/6 for each given turn.

I was under the impression (K_2 | K_1^2) the "|" was specifically used in the context of a dependant outcome.
If this is a dependant probability outcome, shouldn't it have been better explained in the question that an item selected will be removed and landing on it again would require another spin of the board.

unless of course, I have completely misunderstood the "|" given symbol, and is not exclusive to a dependant formulae.
(1 vote)
• Sal does explain this at
If 𝐴 and 𝐵 are independent events, then 𝑃(𝐴 | 𝐵) = 𝑃(𝐴),
because if they are independent events, then the occurrence of 𝐵 does not change the probability of 𝐴.

So, the general multiplication rule applies to dependent events as well as independent events.