Question Video: Applying the Multiplication Rule to Calculate Probabilities Mathematics

For two events 𝐴 and 𝐵, 𝑃(𝐴′) = 0.4 and 𝑃(𝐵′ | 𝐴′) = 0.6. Find 𝑃(𝐴′ ∩ 𝐵′).

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Video Transcript

For two events 𝐴 and 𝐵, the probability of 𝐴 dash, the complement of 𝐴, is equal to 0.4 and the probability of 𝐵 dash given 𝐴 dash is 0.6. Find the probability of 𝐴 dash intersection 𝐵 dash.

The complement of 𝐴 means the probability of 𝐴 not occurring. We recall that this is equal to one minus the probability of 𝐴. The vertical line given that means that we are dealing with conditional probability. The probability that 𝐵 does not occur given that 𝐴 does not occur is 0.6. Another one of our probability formulae states that the probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵. We can use a rearranged version of that to help us work out the probability of 𝐴 dash intersection 𝐵 dash, the probability of 𝐴 not happening and 𝐵 not happening.

Using our conditional probability formula, we have that the probability of 𝐵 dash given 𝐴 dash is equal to the probability of 𝐴 dash intersection 𝐵 dash divided by the probability of 𝐴 dash. Note that the complement of 𝐴 and the complement of 𝐵 on the right-hand side are interchangeable as 𝐴 dash intersection 𝐵 dash is the same as 𝐵 dash intersection 𝐴 dash. We can rearrange this formula by multiplying both sides by the probability of the complement of 𝐴. Substituting in our values gives us 0.4 multiplied by 0.6. This is equal to 0.24. The probability of 𝐴 not occurring and 𝐵 not occurring is 0.24 .

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