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