In a class of students, the following data table summarizes the amount of sleep of students and whether they have an A in the class. What is the probability that a student chosen randomly from the class is a student who slept more than 6 hours and has an A?

[tex]\[
\begin{tabular}{|c|c|c|}
\hline
& \begin{tabular}{c}
Slept more \\
than 6 hours
\end{tabular} & \begin{tabular}{c}
Slept 6 \\
hours or \\
less
\end{tabular} \\
\hline
Has an A & 7 & 2 \\
\hline
\begin{tabular}{c}
Does not \\
have an A
\end{tabular} & 6 & 13 \\
\hline
\end{tabular}
\][/tex]



Answer :

To find the probability that a randomly chosen student from the class is a student who slept more than 6 hours and has an A, we can follow these steps:

1. Identify the specific group of interest and their count: We need the number of students who slept more than 6 hours and have an A. According to the given data table, there are [tex]\(7\)[/tex] such students.

2. Calculate the total number of students in the class: We sum up the number of students across all categories in the table:
- Students who slept more than 6 hours and have an A: [tex]\(7\)[/tex]
- Students who slept 6 hours or less and have an A: [tex]\(2\)[/tex]
- Students who slept more than 6 hours and do not have an A: [tex]\(6\)[/tex]
- Students who slept 6 hours or less and do not have an A: [tex]\(13\)[/tex]

Adding these values gives:
[tex]\[ 7 + 2 + 6 + 13 = 28 \][/tex]

3. Determine the total number of students in the specific group (students who slept more than 6 hours and have an A): This is [tex]\(7\)[/tex], as found in step 1.

4. Compute the probability: The probability [tex]\(P\)[/tex] of a randomly chosen student being from the specific group is given by the ratio of the number of students in that group to the total number of students in the class:
[tex]\[ P(\text{student slept more than 6 hours and has an A}) = \frac{\text{Number of students who slept more than 6 hours and have an A}}{\text{Total number of students}} = \frac{7}{28} \][/tex]

5. Simplify the fraction if possible: Simplifying [tex]\(\frac{7}{28}\)[/tex] gives:
[tex]\[ \frac{7}{28} = \frac{7 \div 7}{28 \div 7} = \frac{1}{4} \][/tex]

Converting this to a decimal form, we get:
[tex]\[ \frac{1}{4} = 0.25 \][/tex]

Thus, the probability that a randomly chosen student from the class is a student who slept more than 6 hours and has an A is [tex]\(0.25\)[/tex] or [tex]\(25\%\)[/tex].