Question

In a class of students, the following data table summarizes how many students have a brother or a sister. What is the probability that a student chosen randomly from the class does not have a brother?
Has a brother Does not have a brother
Has a sister 2 6
Does not have a sister 5 8
Answer

Answer 1

To determine the probability that a randomly chosen student does not have a brother, we first need to calculate the total number of students and the number of students who do not have a brother.

From the table:

- Students who **have a brother** and **have a sister**: 2
- Students who **do not have a brother** and **have a sister**: 6
- Students who **have a brother** and **do not have a sister**: 5
- Students who **do not have a brother** and **do not have a sister**: 8

Let's summarize the data:

- Students who **have a brother**: \( 2 + 5 = 7 \)
- Students who **do not have a brother**: \( 6 + 8 = 14 \)

Now we can calculate the total number of students:

\[
\text{Total Students} = (Has \: brother \: and \: has \: sister) + (Does \: not \: have \: brother \: and \: has \: sister) + (Has \: brother \: and \: does \: not \: have \: sister) + (Does \: not \: have \: brother \: and \: does \: not \: have \: sister)
\]

Substituting the values:

\[
\text{Total Students} = 2 + 6 + 5 + 8 = 21
\]

Next, we focus on the number of students who do not have a brother, which we calculated earlier as 14.

Now we can find the probability that a randomly chosen student does not have a brother:

\[
P(\text{Does not have a brother}) = \frac{\text{Number of students who do not have a brother}}{\text{Total number of students}} = \frac{14}{21}
\]

To simplify this fraction:

\[
\frac{14}{21} = \frac{2}{3}
\]

Thus, the probability that a randomly chosen student does not have a brother is

\[
\boxed{\frac{2}{3}}.
\]