# Video: MATH-STATS-2018-S1-Q01

MATH-STATS-2018-S1-Q01

03:32

### Video Transcript

If the probability of π΄ given π΅ equals one-third and the probability of π΅ equals twelve twenty-fifths, find the probability of π΄ intersect π΅.

Firstly, letβs just clarify some of the notation used in this question. And this first piece here, which is a probability of π΄ and then a vertical line and then π΅, I read this as the probability of π΄ given π΅. And it just means the conditional probability of the event π΄ occurring given that we already know the event π΅ has occurred.

Secondly, this piece of notation here, which I read as the probability of π΄ intersect π΅. On a Venn diagram, the intersection of two sets π΄ and π΅ is their overlap. So, when we talk about the probability of π΄ intersect π΅, we mean the probability that events π΄ and π΅ both happen.

We have a formula that connects the three probabilities that weβre given in the question. The probability of π΄ and π΅ both occurring can be found by finding the probability that π΅ occurs and then multiply it by the probability of π΄ given π΅. Thatβs the probability that π΄ occurs given that π΅ has already occurred.

Dividing both sides of this formula through by the probability of π΅ gives a rearrangement. The probability of π΄ given π΅ is equal to the probability of π΄ intersect π΅ divided by the probability of π΅. And this is called the conditional probability formula. And it gives us a definition for the probability of π΄ given π΅.

However, in this question, we are asked to find the probability of π΄ intersect π΅. So, we can use the first version of this formula that we wrote down. The probability of π΅ is twelve twenty-fifths. And the probability of π΄ given π΅ is one-third. So, we have that the probability of π΄ intersect π΅ is twelve twenty-fifths multiplied by one-third. We can simplify these fractions by cross-cancelling a factor of three from the 12 in the numerator of the first fraction and the three in the denominator of the second to give four twenty-fifths multiplied by one over one. This is just equal to four twenty-fifths. So, the probability of π΄ intersect π΅ for the events with the given probabilities is four twenty-fifths.

Now just a more general comment, before you learned about conditional probability, you probably learned that if events π΄ and π΅ are independent, then to find the probability of them both happening, so thatβs the probability of their intersection, we can multiply the individual probabilities together. So, we just multiply the probability of π΅ by the probability of π΄.

The conditional probability formula is really just an extension of this, to situations where the events π΄ and π΅ are not independent. We still have the probability of π΅, but we replace the probability of π΄ with the probability of π΄ given π΅. Because the probability for the second event will be affected by the outcome of the first.

At any rate, if the events π΄ and π΅ are independent, then the probability of π΄ given π΅ will just be equal to the probability of π΄. So, our conditional probability formula is just a more general version of the formula we use when the events are independent. We found that, in this question, the probability of π΄ intersect π΅ is four twenty-fifths. This fraction canβt be simplified any further, as the numerator and denominator have no common factors other than one.