Question Video: Determining the Probability of an Event given the Difference and the Intersection of Two Events Mathematics

Suppose that 𝐴 and 𝐵 are two events. Given that P(𝐴 − 𝐵) = 2/7 and P(𝐴 ∩ 𝐵) = 1/6, determine P(𝐴).

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Video Transcript

Suppose that 𝐴 and 𝐵 are two events. Given that the probability of 𝐴 minus 𝐵 is equal to two-sevenths and the probability of 𝐴 intersection 𝐵 is one-sixth, determine the probability of 𝐴.

We begin this question by recalling the difference rule for probability. This states that the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. We are told that the probability of 𝐴 minus 𝐵 is equal to two-sevenths and the probability of 𝐴 intersection 𝐵 is one-sixth. Substituting these values into our formula, we have two-sevenths is equal to the probability of 𝐴 minus one-sixth. In order to calculate the probability of 𝐴, we can add one-sixth to both sides of this equation. The probability of 𝐴 is equal to two-sevenths plus one-sixth.

In order to add any two fractions, we begin by finding a common denominator. In this case, we’ll use 42 as this is the lowest common multiple of seven and six. Multiplying the numerator and denominator of our first fraction by six gives us 12 over 42, and multiplying the numerator and denominator of the second fraction by seven gives us seven over 42. Two-sevenths plus one-sixth is therefore equal to 19 over 42. We can therefore conclude that the probability of event 𝐴 is 19 over 42.

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