Question Video: Calculating Conditional Probability to Decide Whether Events Are Independent | Nagwa Question Video: Calculating Conditional Probability to Decide Whether Events Are Independent | Nagwa

# Question Video: Calculating Conditional Probability to Decide Whether Events Are Independent Mathematics • Third Year of Secondary School

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Suppose P(π΄) = 2/5 and P(π΅) = 3/7. The probability that event π΄ occurs and event π΅ also occurs is 1/5. Calculate P(π΄ | π΅), and then evaluate whether events π΄ and π΅ are independent.

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### Video Transcript

Suppose the probability of π΄ is two-fifths and the probability of π΅ is three-sevenths. The probability that event π΄ occurs and event π΅ also occurs is one-fifth. Calculate the probability of π΄ given π΅ and then evaluate whether events π΄ and π΅ are independent.

Well, first, let us recall the conditional probability formula. The probability of π΄ given π΅ is equal to the probability of π΄ intersection π΅ over the probability of π΅. Well, the probability of π΄ intersection π΅ is the probability that event π΄ occurs and event π΅ also occurs, which we were given in the question is a fifth. And we were also told in the question that the probability of event π΅ occurring is three-sevenths. So the probability of π΄ given π΅ is a fifth divided by three-sevenths. And, of course, an equivalent operation to dividing by a fraction is multiplying by the reciprocal of that fraction. So this is equal to a fifth times seven over three. Well, thereβs the answer to our first part of the question. The probability of π΄ given π΅ is equal to seven-fifteenths.

For the second part of the question, letβs recall our test for independence using the conditional probability formulae. If the probability of π΄ is equal to the probability of π΄ given π΅ and the probability of π΅ is equal to the probability of π΅ given π΄, then we can say that events π΄ and π΅ are independent. Now the question told us that the probability of π΄ is two-fifths. So the next step is to work out the probability of π΄ given π΅. Well, in fact, thatβs what we worked out in the first part of the question. So we know that the probability of π΄ given π΅ is seven-fifteenths.

Now, if two-fifths is the same as seven-fifteenths, then weβd need to go on to check whether the probability of π΅ is equal to the probability of π΅ given π΄. Now to compare two-fifths and seven-fifteenths, we need to get a common denominator. We can do this by multiplying the numerator and denominator by three and two-fifths becomes six-fifteenths. That means that the probability of π΄ is not equal to the probability of π΄ given π΅. And the fact that we need both of these conditions to be true to prove the independence of events π΄ and π΅, the fact that weβve shown that the probability of π΄ is not equal to the probability of π΄ given π΅, means that we know that these two events arenβt independent. We donβt even need to go on to check whether the probability of π΅ is equal to the probability of π΅ given π΄.

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