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.