Question Video: Using the Addition Rule in Probability Mathematics

For two events 𝐴 and 𝐵, 𝑃(𝐴) = 3/5, 𝑃(𝐵) = 3/4, and 𝑃(𝐴 ∪ 𝐵) = 5/6. Work out the probability of 𝐴 ∩ 𝐵.

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Video Transcript

For two events 𝐴 and 𝐵, the probability of 𝐴 is three-fifths, the probability of 𝐵 is three-quarters, and the probability of 𝐴 union 𝐵 is five-sixths. Work out the probability of 𝐴 intersection 𝐵.

In order to answer this question, we will use the addition rule of probability. This states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. In this question, we are told the probability of 𝐴 is three-fifths, the probability of 𝐵 is three-quarters, and the probability of 𝐴 union 𝐵 is five-sixths. We are asked to work out the probability of 𝐴 intersection 𝐵.

We can rearrange this equation by adding the probability of 𝐴 intersection 𝐵 and subtracting five-sixths from both sides. The probability of 𝐴 intersection 𝐵 is therefore equal to three-fifths plus three-quarters minus five-sixths. In order to add and subtract fractions, we begin by finding a common denominator. In this case, we need to find the lowest common multiple of five, four, and six. This is equal to 60, as 60 is the lowest number in the four, five, and six times table.

Multiplying the numerator and denominator of our first fraction by 12 gives us 36 over 60. As four multiplied by 15 is 60, we can multiply the numerator and denominator of the second fraction by 15 to give us 45 over 60. By multiplying the numerator and denominator of the third fraction by 10, we see that five over six is equivalent to 50 over 60. As all three fractions now have a denominator of 60, we can simply add and then subtract the numerators. 36 plus 45 is equal to 81. And subtracting 50 gives us 31.

If the probability of 𝐴 is three-fifths; the probability of 𝐵, three-quarters; the probability of 𝐴 union 𝐵, five-sixths, then the probability of 𝐴 intersection 𝐵 using the addition rule of probability is 31 over 60 or thirty-one sixtieths.

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