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Course: Precalculus > Unit 8
Lesson 2: Multiplication rule for probabilities- Compound probability of independent events
- Independent events example: test taking
- General multiplication rule example: independent events
- Dependent probability introduction
- General multiplication rule example: dependent events
- Probability with general multiplication rule
- Interpreting general multiplication rule
- Interpret probabilities of compound events
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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?(6 votes)
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!(6 votes)
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.(5 votes)
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.(2 votes)
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!(5 votes)
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!(0 votes)
- I am also confused about this.
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 at1:32
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.(2 votes)
1:32Video transcript
- [Instructor] We're
told that two contestants are finalists in a cooking competition. For the final round,
each of them spin a wheel to determine what star
ingredient must be in their dish. I guess the primary ingredient, and we can see it could be chard, spinach, romaine lettuce, I'm guessing,
cabbage, arugula, or kale. And so then they give us these
different types of events, or at least the symbols for
these different types of events, and then give us their meaning. So K-sub 1 means, the first
contestant lands on kale, K-sub 2 means, the second
contestant lands on kale, K-sub 1 with this superscript C, which you could view as complement. So K-sub 1 one complement, the first contestant
does not land on kale. So it's the complement of
this one right over here. And then K-sub 2 complement, would be that the second
contestant does not land on kale. So the not of K-sub 2 right over here. Using the general multiplication rule, express symbolically the probability that neither contestant lands on kale. So pause this video and see
if you can have a go at this. All right, so the general
multiplication rule is just saying this notion that the probability
of two events, A and B, is going to be equal
to the probability of, let's say A given B, times
the probability of B. Now, if they're independent events, if the probability of A occurring does not depend in any way
on whether B occurred or not, then this would simplify to
this probability of A given B, would just become the probability of A. And so if you have two independent events, you would just multiply their probability. So that's just all they're talking about, the general multiplication rule. But let me express what they're actually
asking us to express. The probability that neither
contestant lands on kale. So that means that this
is going to happen, the first contestant
does not land on kale, and this is going to happen, the second contestant
does not land on kale. So I could write it this way. The probability that K-sub 1 complement and K-sub 2 complement, and
I could write it this way. This is going to be equal to, we know that these are independent events because if the first contestant gets kale or whatever they get it, it doesn't get taken out of the running for the second contestant. The second contestant still
has an equal probability of getting or not getting kale, regardless of what happened
for the first contestant. So that means we're just in the situation where we multiply these probabilities. So that's gonna be the
probability of K-sub 1 complement, times the probability
of K-sub 2 complement. All right, now let's do part two. Interpret what each part of this probability statement represents. So I encourage you like always, pause this video and
try to figure that out. All right, so first let's think
about what is going on here. So this is saying, the probability that this
is K-sub 1 complement. So the first contestant
does not land on kale. So first, first contestant
does not get kale, and, I'll write and in caps, and second contestant does get kale. And second does get kale. So that's what this left-hand is saying. And now they say that that
is going to be equal to, so this part right over here, probability that the first
contestant does not get kale. Probability that first does not get kale, times, right over here. And the second part right over here is the probability that the
second contestant gets kale, given that the first
contestant does not get kale. So probability that the second gets kale, given, that's what this vertical
line right over here means, it means given, shorthand for given. Given, I wrote it up there too. Given that first does not get kale. And we're done, we've just
explained what is going on here.