Video: Calculating the Conditional Probability of Two Independent Events

For two independent events 𝐴 and 𝐵, where 𝑃(𝐴) = 0.45 and the 𝑃(𝐵) = 0.3, calculate 𝑃(𝐴 | 𝐵′).

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Video Transcript

For two independent events 𝐴 and 𝐵, where the probability of 𝐴 is 0.45 and the probability of 𝐵 is 0.3, calculate the probability of 𝐴 given the complement of 𝐵.

The vertical line in our question means given that. The hash mark means the complement. This means the probability of 𝐵 not occurring. In order to answer this question, we need to recall some of our probability formulae. The probability of the complement of 𝐴 is equal to one minus the probability of 𝐴. The probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 divided by the probability of 𝐵. We are also told in this question that our events are independent. For independent events, the probability of 𝐴 intersection 𝐵 is equal to the probability of 𝐴 multiplied by the probability of 𝐵.

As the probability of 𝐵 is 0.3, the complement of this will be one minus 0.3. The probability of 𝐵 not occuring is 0.7. As our events are independent, the probability of 𝐴 occurring and 𝐵 not occurring is equal to 0.45 multiplied by 0.7. We multiply the probability of 𝐴 by the probability of the complement of 𝐵. This is equal to 0.315. We can, therefore, calculate the probability of 𝐴 occurring given that 𝐵 does not occur by dividing 0.315 by 0.7. We divide the probability of 𝐴 occurring and 𝐵 not occurring by the probability of 𝐵 not occurring. This is equal to 0.45.

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