Question Video: Applying the Multiplication Rule to Calculate Probabilities

For two events 𝐴 and 𝐡, P(𝐡′) = 0.3 and P(𝐴 ∣ 𝐡) = 0.3. Determine the probability of 𝐴 ∩ 𝐡

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Video Transcript

For two events 𝐴 and 𝐡, the probability of 𝐡 prime equals 0.3 and the probability of 𝐴 given 𝐡 equals 0.3. Determine the probability of 𝐴 intersection 𝐡.

The first thing to note in this question is we’re not given the probability of 𝐡, but instead we’re given the probability of 𝐡 prime. This refers to the complement of 𝐡, and it’s essentially the probability of 𝐡 not occurring. As we know all probabilities sum to one, then this means that the probability of 𝐡 is equal to one subtract 0.3, and that’s 0.7. We’re given the probability of 𝐴 given 𝐡 as 0.3, and we need to find the probability of 𝐴 intersection 𝐡.

We can use the formula for conditional probability that tells us that the probability of 𝐴 given 𝐡 is equal to the probability of 𝐴 intersection 𝐡 divided by the probability of 𝐡. As we want to find the probability of 𝐴 intersection 𝐡, we can rearrange this formula to help us. Plugging the values into the formula, the probability of 𝐴 given 𝐡 is 0.3 and the probability of 𝐡 is 0.7, so we multiply those together, which gives us the answer that the probability of 𝐴 intersection 𝐡 is 0.21.

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