Question

Out of 80 customers at an ice cream van, 48 had syrup, 28 had sprinkles and 16 had both

toppings on their ice cream. Use a Venn diagram to find the probability that a randomly
selected customer doesn't have either topping, given that they don't have sprinkles.

I know the answer is 20/52, I just can’t work out how to get to that answer…

Answer 1

20/52 or simplified to 5/13

Explanation:

The Venn Diagram is provided

Let
n(A) = number of customers who had syrup
n(B) = number of customers who had sprinkles

n(A and B) = number of customers who had both syrup and sprinkles = 16
This would be the number in the overlapping region

n(A or B) = number of customers who had either syrup or sprinkles or both
= n(A) + n(B) - n(A and B)
= 48 + 28 - 16
= 60

Therefore number of customers who had neither topping = 80 - 60 = 20

This number is indicated outside both circles but within the rectangle

The number of customers who had only syrup is given by set difference

= No. of customers who had syrup - No. of customers who had both
= n(A) - n(A and B)
= 48 - 16
= 32

This is the figure inside the left circle

Let's consider the statement: Customers who didn't have sprinkles

This would be customers who had only syrup(32) + customers who had neither topping(20)
= 32 + 20 = 52

Number of customers who did not have either topping = 20
P(selected customer doesn't have either topping, given that they don't have sprinkles)
= 20/52
= 5/13