Question

Helen has a box of marbles.

1
2
of the marbles are yellow.
1
8
of the marbles are red.
The rest of the marbles are blue.
Helen pulls one marble out of the box at random, records its color, replaces it, and mixes up the marbles again. If she does this 200 times, how many blue marbles should she expect to pull out?

A.
104

B.
120

C.
67

D.
75

Answer 1

Let's denote the total number of marbles in the box as \( N \).

From the problem, we know:
- The fraction of yellow marbles is \( \frac{1}{2} \).
- The fraction of red marbles is \( \frac{1}{8} \).

To find the fraction of blue marbles, we first need to calculate the combined fraction of yellow and red marbles:

\[
\text{Fraction of yellow and red marbles} = \frac{1}{2} + \frac{1}{8}
\]

To add these fractions, we first find a common denominator. The least common multiple of 2 and 8 is 8.

Converting \( \frac{1}{2} \) to have a denominator of 8:

\[
\frac{1}{2} = \frac{4}{8}
\]

Now, we can add the fractions:

\[
\frac{4}{8} + \frac{1}{8} = \frac{5}{8}
\]

This means that \( \frac{5}{8} \) of the marbles are either yellow or red. Therefore, the fraction of blue marbles is:

\[
\text{Fraction of blue marbles} = 1 - \frac{5}{8} = \frac{3}{8}
\]

Now, if Helen performs the experiment of pulling out a marble 200 times, the expected number of blue marbles pulled out is given by:

\[
\text{Expected number of blue marbles} = 200 \times \frac{3}{8}
\]

Calculating this gives:

\[
200 \times \frac{3}{8} = \frac{600}{8} = 75
\]

Thus, the expected number of blue marbles that Helen should expect to pull out is:

\[
\boxed{75}
\]