Video: Finding Conditional Probabilities

Suppose 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴) = 0.75 and 𝑃(𝐡) is 0.5. Given that 𝑃(𝐴 ∩ 𝐡) = 0.44, find the probability that 𝐡 occurs given that 𝐴 does not.

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Video Transcript

Suppose 𝐴 and 𝐡 are events with probabilities. The probability of 𝐴 is 0.75 and probability of 𝐡 is 0.5. Given that the probability of 𝐴 intersection 𝐡, 𝐴 and 𝐡 both occurring, is 0.44, find the probability that 𝐡 occurs given that 𝐴 does not.

The words β€œgiven that” means that we are dealing with conditional probability. We recall that the probability of 𝐴 given 𝐡 is equal to the probability of 𝐴 intersection 𝐡, the probability of both events occurring, divided by the probability of 𝐡. We also recall that the probability of 𝐴 not occurring, the complement of 𝐴, is equal to one minus the probability of 𝐴. In this question, we want to calculate the probability of 𝐡 occurring, given that 𝐴 does not occur. This is equal to the probability of 𝐡 occurring and 𝐴 not occurring divided by the probability of 𝐴 not occurring.

The probability of 𝐴 not occurring is one minus 0.75 as the probability of 𝐴 occurring was 0.75. One minus 0.75 is 0.25. This means that our denominator will be 0.25. There is also a formula that helps us calculate the probability of 𝐡 occurring and 𝐴 not occurring. This is equal to the probability of 𝐡 minus the probability of 𝐴 intersection 𝐡. We are told both of these values in the question. They are 0.5 and 0.44. The probability of 𝐡 occurring and 𝐴 not occurring is therefore equal to 0.5 minus 0.44. This is equal to 0.06. This means that we need to divide this by 0.25. 0.06 divided by 0.25 is 0.24. The probability that 𝐡 occurs given that 𝐴 does not is 0.24.

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