Video Transcript
Suppose 𝐴 and 𝐵 are events. Given that the probability of 𝐴 is four 𝑥, the probability of the complement of 𝐵 is 𝑥, the probability of 𝐴 union 𝐵 is three 𝑥 plus 0.9, and the probability of 𝐴 intersection 𝐵 is a half 𝑥, find the value of 𝑥.
Let’s begin by recalling some of the notation in this question. The complement of an event is the set of elements not in that event. This can be written as in this case as 𝐵 bar and is also sometimes written 𝐵 prime. As probabilities sum to one, we know that the probability of the complement of event 𝐵 is equal to one minus the probability of event 𝐵. In this question, we can rearrange this equation to find an expression for the probability of 𝐵. This is equal to one minus 𝑥.
The union notation gives us the set of elements in event 𝐴 or event 𝐵 or in both. And the intersection notation represents those elements that are in event 𝐴 and in event 𝐵. The addition rule of probability 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. We are told that the probability of 𝐴 union 𝐵 is three 𝑥 plus 0.9. The probability of 𝐴 is four 𝑥. We have worked out that the probability of event 𝐵 is one minus 𝑥. And finally, the probability of 𝐴 intersection 𝐵 is a half 𝑥, which gives us the equation shown.
Collecting like terms on the right-hand side, we have 2.5𝑥 plus one, as four 𝑥 minus 𝑥 minus a half 𝑥 is 2.5𝑥. Next, we can subtract 2.5𝑥 and 0.9 from both sides of our equation. This gives us three 𝑥 minus 2.5𝑥 is equal to one minus 0.9. And this in turn simplifies to 0.5𝑥 is equal to 0.1. Dividing through by 0.5 or a half, we have 𝑥 is equal to 0.1 divided by 0.5. And multiplying the numerator and denominator of this fraction by 10 gives us one over five or one-fifth. The value of 𝑥 is one-fifth or, written as a decimal, 0.2.
