Questions: At a coffee shop, the first 100 customers' orders were as follows. Small Medium Large Total --------------------------------- Hot 5 48 22 75 Cold 8 12 5 25 Total 13 60 27 100 Find the probability a customer ordered a small, given that they ordered a cold drink. P(small cold) = P(small and cold) / P(cold) = [?]

At a coffee shop, the first 100 customers' orders were as follows.

Small Medium Large Total
---------------------------------
Hot 5 48 22 75
Cold 8 12 5 25
Total 13 60 27 100

Find the probability a customer ordered a small, given that they ordered a cold drink.

P(small cold) = P(small and cold) / P(cold) = [?]

Transcript text: At a coffee shop, the first 100 customers' orders were as follows. \begin{tabular}{|c|c|c|c|c|} \hline & Small & Medium & Large & Total \\ \hline Hot & 5 & 48 & 22 & 75 \\ \hline Cold & 8 & 12 & 5 & 25 \\ \hline Total & 13 & 60 & 27 & 100 \\ \hline \end{tabular} Find the probability a customer ordered a small, given that they ordered a cold drink. \[ P(\text { small } \mid \text { cold })=\frac{P(\text { small and cold })}{P(\text { cold })}=[?] \]

Solution

Solution Steps

Step 1: Identify the Events

Let's denote \(A\) as the event of interest and \(B\) as the condition event. The goal is to find the probability of \(A\) occurring given that \(B\) has occurred.

Step 2: Calculate \(P(A \text{ and } B)\)

The joint probability of events \(A\) and \(B\) occurring together is given as \(P(A \text{ and } B) = 8\).

Step 3: Calculate \(P(B)\)

The probability of event \(B\) occurring is given as \(P(B) = 25\).

Step 4: Apply the Formula for Conditional Probability

The formula for conditional probability is \[P(A|B) = \frac{P(A \text{ and } B)}{P(B)}\]. Substituting the given values, we get \[P(A|B) = \frac{8}{25} = 0.32\].