Question

There are 16 marbles in a jar: 5 red, 3 blue, and 8 yellow. What is the probability of selecting a blue marble and then, without replacing the blue marble, selecting a yellow marble?

A. 3/32
B. 11/16
C. 173/240
D. 1/10

Answer 1

To find the probability of selecting a blue marble first and then a yellow marble, we can break it down into two steps.

### Step 1: Selecting a blue marble
There are 3 blue marbles out of a total of 16 marbles.

The probability of selecting a blue marble is:
\[
P(\text{Blue}) = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{3}{16}
\]

### Step 2: Selecting a yellow marble after removing the blue marble
After removing a blue marble from the jar, there are now 15 marbles left (because we did not replace the blue marble). The composition of the remaining marbles is:
- 5 red marbles
- 3 blue marbles (originally, but one has been taken out)
- 8 yellow marbles

So the total number of yellow marbles remains 8.

The probability of selecting a yellow marble now is:
\[
P(\text{Yellow | Blue}) = \frac{\text{Number of yellow marbles}}{\text{Total number of remaining marbles}} = \frac{8}{15}
\]

### Final Calculation: Combined Probability
Now, we multiply the probabilities of the two independent events to find the total probability of selecting a blue marble and then a yellow marble.

\[
P(\text{Blue and then Yellow}) = P(\text{Blue}) \times P(\text{Yellow | Blue}) = \frac{3}{16} \times \frac{8}{15}
\]

Calculating that:

\[
P(\text{Blue and then Yellow}) = \frac{3 \times 8}{16 \times 15} = \frac{24}{240} = \frac{1}{10}
\]

Thus, the probability of selecting a blue marble and then, without replacing it, selecting a yellow marble is:

\[
\boxed{\frac{1}{10}}
\]