To solve the problem, we need to find the conditional probability that a customer ordered a hot drink given that they ordered a large drink. This is represented as [tex]\( P(\text{hot} \mid \text{large}) \)[/tex].
The formula for conditional probability is:
[tex]\[
P(\text{hot} \mid \text{large}) = \frac{P(\text{hot and large})}{P(\text{large})}
\][/tex]
Here is a step-by-step breakdown of how to find this probability:
1. Identify [tex]\( P(\text{hot and large}) \)[/tex]:
From the table, we can see that the number of customers who ordered a hot large drink is 22. Since the total number of customers is 100, the probability that a customer ordered a hot large drink is:
[tex]\[
P(\text{hot and large}) = \frac{\text{Number of hot large orders}}{\text{Total number of customers}} = \frac{22}{100}
\][/tex]
2. Identify [tex]\( P(\text{large}) \)[/tex]:
The total number of large drink orders, both hot and cold, is 27. Therefore, the probability that a customer ordered a large drink is:
[tex]\[
P(\text{large}) = \frac{\text{Number of large orders}}{\text{Total number of customers}} = \frac{27}{100}
\][/tex]
3. Calculate the conditional probability:
Using the values identified above, we plug them into the formula for conditional probability:
[tex]\[
P(\text{hot} \mid \text{large}) = \frac{P(\text{hot and large})}{P(\text{large})} = \frac{\frac{22}{100}}{\frac{27}{100}} = \frac{22}{27}
\][/tex]
4. Convert the fraction to a decimal:
To express the probability as a decimal, we divide:
[tex]\[
\frac{22}{27} \approx 0.8148148148148148
\][/tex]
So, the probability that a customer ordered a hot drink given that they ordered a large drink is approximately [tex]\( 0.8148 \)[/tex] or [tex]\( 81.48\% \)[/tex].
