A survey was conducted on a group of 263 children to determine their favorite superheroes. The results are shown in the Venn diagram. Find the probability of Wonder Woman or Batman.
We are told that there were a total of 263 children that took part in a survey. In response to the question “Who is your favorite superhero?,” they responded Wonder Woman, Batman, Superman, or a combination of all three. In this question, we are interested in the children who said Wonder Woman or Batman. We need to calculate the probability that they answered at least one of these superheroes. This is known as the union of the two events.
One way of calculating this is to simply find the sum of all the values inside the circle for Wonder Woman or the circle for Batman. 64 children answered just Wonder Woman. Eight children answered Wonder Woman and Superman. Nine children answered Wonder Woman and Batman. And eight children answered all three of the superheroes.
We note that these last two values are also inside the circle for Batman. As we don’t want to count these children twice, we just need to add 14 and 75. There were 75 children who said just Batman and 14 children said Batman and Superman. The sum of these six values is 178. This means that the probability that a child said Wonder Woman or Batman is 178 out of 263. An alternative method here would be to consider those children who only said Superman. There were 85 of these. So we can subtract 85 from 263 to once again give us the 178 children that said either Wonder Woman or Batman.
Whilst it was not required in this question, we could’ve used the addition rule of probability to solve the problem. 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. If we let 𝐴 be the event that a child said Wonder Woman, then the probability of 𝐴 is 89 over 263, as the numbers inside the circle representing Wonder Woman sum to 89. If 𝐵 is the event that a child said Batman, then the probability of event 𝐵 is 106 over 263.
As already mentioned, there were 17 children that answered Batman and Wonder Woman. Therefore, the probability of 𝐴 intersection 𝐵 is 17 over 263. The probability of 𝐴 union 𝐵 is therefore equal to 89 over 263 plus 106 over 263 minus 17 over 263. This is once again equal to 178 over 263. This is the probability a child said Wonder Woman or Batman.