Two mutually exclusive events 𝐴 and 𝐵 have probabilities the probability of 𝐴 equals one-tenth and the probability of 𝐵 equals one-fifth. Find the probability of 𝐴 union 𝐵.
We begin by recalling the definition of mutually exclusive events. Two or more events are said to be mutually exclusive if they cannot happen at the same time. This can be represented on a Venn diagram as shown. As there is no overlap between the two circles, the probability of 𝐴 intersection 𝐵 is zero. We know that the probability of 𝐴 union 𝐵 is the probability of 𝐴 or 𝐵 or both occurring. However, in this case, they cannot occur together. Therefore, the probability of 𝐴 union 𝐵 for mutually exclusive events is the probability of 𝐴 plus the probability of 𝐵.
These two rules are true for all mutually exclusive events. We are told that the probability of 𝐴 in this question is one-tenth. The probability of 𝐵 is one-fifth. This means that the probability of 𝐴 union 𝐵 is equal to one-tenth plus one-fifth. The fraction one-fifth is equivalent to two-tenths. We can multiply the numerator and denominator by two. As one-tenth plus two-tenths is three-tenths, this is the probability of 𝐴 union 𝐵.
Whilst it is not required in this question, we can complete the Venn diagram by calculating the probability that both events 𝐴 and 𝐵 do not happen. As probability sum to one, this is equal to one minus three-tenths which is equal to seven-tenths. On the Venn diagram, the fraction seven-tenths will lie outside of circle 𝐴 and outside of circle 𝐵 as shown.