Question

an ice - cream shop tracks the orders for different flavors and styles of ice - cream to stock the best inventory. the two - way frequency table summarizes the orders.

	chocolate	vanilla	strawberry	mint chocolate
soft serve	10	15	12	8
sundae	27	43	10	5
milkshake	33	17	5	10

in the context of the data, interpret the empirical conditional probability of 43/85.
the probability that a customer ordered vanilla ice - cream.
the probability that a customer ordered a sundae made of vanilla ice - cream.
the probability that a customer ordered a sundae, given that they ordered vanilla ice - cream.
the probability that a customer ordered vanilla ice - cream, given that they ordered a sundae.

Explanation:

Step1: Identify relevant values

We want to find the probability that a customer ordered vanilla ice - cream given that they ordered a sundae. The number of customers who ordered vanilla sundaes is 43, and the total number of customers who ordered sundaes is \(5 + 10+43 + 27=85\).

Step2: Apply conditional - probability formula

The formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). In terms of frequencies, if \(A\) is the event of ordering vanilla ice - cream and \(B\) is the event of ordering a sundae, the probability \(P(A|B)=\frac{n(A\cap B)}{n(B)}\), where \(n(A\cap B)\) is the number of elements in the intersection of \(A\) and \(B\), and \(n(B)\) is the number of elements in \(B\). Here, \(n(A\cap B) = 43\) and \(n(B)=85\).

Answer:

The probability that a customer ordered vanilla ice - cream, given that they ordered a sundae is \(\frac{43}{85}\).