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 hot drink, given that they ordered a large.

[tex]
P (\text { hot } \mid \text { large })=\frac{ P (\text { hot and large })}{ P (\text { large })}=[?]
[/tex]



Answer :

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].