Question

Mr. Blackwells's class is conducting an experiment to find the probability of pulling certain colors from a bag of 25 marbles. If 7 are purple, 4 are yellow, 5 are green, and the rest are black, what is the probability of drawing a purple and yellow if the marbles are not replaced after they are picked?
A.

B.

C.

D.

Answer 1

To calculate the probability of drawing a purple marble and then a yellow marble from the bag of 25 marbles without replacement, we first need to determine the total number of each color of marble:

- Purple: 7
- Yellow: 4
- Green: 5
- Black: \( 25 - (7 + 4 + 5) = 25 - 16 = 9 \)

Now we have:

- Total marbles = 25
- Purple marbles = 7
- Yellow marbles = 4

**Step 1: Calculate the probability of drawing a purple marble first.**

The probability of drawing a purple marble first is given by the formula:

\[
P(\text{Purple}) = \frac{\text{Number of purple marbles}}{\text{Total number of marbles}} = \frac{7}{25}
\]

**Step 2: Calculate the probability of drawing a yellow marble second (after drawing a purple marble).**

After drawing a purple marble, there are now 24 marbles left in total, and the number of yellow marbles remains at 4. Thus, the probability of drawing a yellow marble after drawing a purple marble is:

\[
P(\text{Yellow | Purple}) = \frac{\text{Number of yellow marbles}}{\text{Total number of remaining marbles}} = \frac{4}{24} = \frac{1}{6}
\]

**Step 3: Calculate the combined probability of both events (drawing purple first and then yellow).**

To find the combined probability of both events occurring, we multiply the probabilities from Step 1 and Step 2:

\[
P(\text{Purple and then Yellow}) = P(\text{Purple}) \times P(\text{Yellow | Purple}) = \frac{7}{25} \times \frac{1}{6}
\]

Now we calculate:

\[
P(\text{Purple and then Yellow}) = \frac{7 \times 1}{25 \times 6} = \frac{7}{150}
\]

Thus, the final probability of drawing a purple marble and then a yellow marble, without replacement, is \(\frac{7}{150}\).

So the answer is \(\frac{7}{150}\).

If you have multiple choice options (A, B, C, D), you can match this answer with the correct letter.