Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other | Nagwa Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other | Nagwa

Question Video: Finding the Probability of Two Events Occuring Together Using the Conditional Probability of One given the Other Mathematics • Third Year of Secondary School

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Suppose 𝑃(𝐡 | 𝐴) = 1/2 and 𝑃(𝐴) = 3/7. What is the probability that events 𝐴 and 𝐡 both occur?

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Video Transcript

Suppose the probability of 𝐡 given 𝐴 equals a half and the probability of 𝐴 equals three-sevenths. What is the probability that events 𝐴 and 𝐡 both occur?

Well, what this problem is, is a conditional probability problem. And that is because we’ve got a condition here. We’re looking at the probability that 𝐡 occurs given that 𝐴 occurs. So therefore, what we’re gonna do is use one of the formulae we have to actually solve conditional probability problems. And the formula that we’re going to use is this one. The probability of 𝐡 given 𝐴 is equal to the probability of 𝐴 intersection 𝐡 divided by the probability of 𝐴.

Well, if we look at what we want to find in this problem, we want to find out the probability that the events 𝐴 and 𝐡 both occur. But what this is, is the property of 𝐴 and 𝐡. And this is the same as the probability of 𝐴 intersection 𝐡. And this is what we’ve got here. So all we need to do now is use some algebra to rearrange our formula. And if we rearrange the formula to make the probability of 𝐴 intersection 𝐡 our subject, then what we’re gonna get is the probability of 𝐴 intersection 𝐡 is equal to the probability of 𝐡 given 𝐴 multiplied by the probability of 𝐴.

So therefore, the probability of 𝐴 intersection 𝐡 is going to be equal to a half multiplied by three-sevenths. And if we remind ourselves how we multiply fractions, what we do is multiply the numerators and multiply the denominators. Well, this is gonna be equal to three fourteenths. So therefore, we can say that if we suppose the probability of 𝐡 given 𝐴 is equal to a half and the probability of 𝐴 is equal three-sevenths, then the probability that events 𝐴 and 𝐡 both occur is three fourteenths.

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