Video Transcript
𝐴 and 𝐵 are two events in a sample space of a random experiment, where the probability of 𝐴 is three-tenths, the probability of 𝐵 is one-fifth, and the probability of 𝐴 minus 𝐵 is one-tenth. Find the probability of 𝐴 union 𝐵.
In this question, we are asked to find the probability of 𝐴 union 𝐵. This is the probability that event 𝐴 or event 𝐵 or both occur. And 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. In this question, we are given values for the probability of 𝐴 and the probability of 𝐵. However, we are not told the probability of 𝐴 intersection 𝐵, where this is the probability that both 𝐴 and 𝐵 occur.
We are told that the probability of 𝐴 minus 𝐵 is one-tenth. This is the probability that 𝐴 occurs but 𝐵 does not as shown on the Venn diagram. This can be calculated by subtracting the probability of 𝐴 intersection 𝐵 from the probability of 𝐴. Substituting the values we know into this equation, we have one-tenth is equal to three-tenths minus the probability of 𝐴 intersection 𝐵. This can be rearranged so that the probability of 𝐴 intersection 𝐵 is equal to three-tenths minus one-tenth, which is equal to two-tenths or one-fifth.
We now have the three values that enable us to calculate the probability of 𝐴 union 𝐵. This is equal to three-tenths plus one-fifth minus one-fifth. As the one-fifths cancel, we are just left with three-tenths or 0.3. If the probability of 𝐴 is three-tenths, the probability of 𝐵 is one-fifth, and the probability of 𝐴 minus 𝐵 is one-tenth, then the probability of 𝐴 union 𝐵 is three-tenths.
