Question
Consider a jar that has 20 marbles. There are 4 red, 4 blue, 4 green, 4 yellow, and 4 purple marbles. What is the probability of selecting a yellow marble without replacing it and then selecting a purple marble? (1 point)
Answers
GPT-4o mini
To find the probability of selecting a yellow marble first and then a purple marble without replacement, follow these steps:
1. **Determine the total number of marbles**:
There are 20 marbles in total: 4 red, 4 blue, 4 green, 4 yellow, and 4 purple.
2. **Calculate the probability of selecting a yellow marble first**:
The number of yellow marbles is 4, so the probability of selecting a yellow marble is:
\[
P(\text{yellow}) = \frac{\text{Number of yellow marbles}}{\text{Total number of marbles}} = \frac{4}{20} = \frac{1}{5}
\]
3. **Determine the situation after selecting a yellow marble**:
After selecting a yellow marble, the total number of marbles left is 19, and the number of purple marbles remains 4.
4. **Calculate the probability of selecting a purple marble next**:
Now, the probability of selecting a purple marble after having removed one yellow marble is:
\[
P(\text{purple | yellow}) = \frac{\text{Number of purple marbles}}{\text{Total number of remaining marbles}} = \frac{4}{19}
\]
5. **Calculate the total probability of both events occurring**:
The total probability of both selecting a yellow marble and then selecting a purple marble is the product of the two probabilities:
\[
P(\text{yellow then purple}) = P(\text{yellow}) \times P(\text{purple | yellow}) = \left(\frac{1}{5}\right) \times \left(\frac{4}{19}\right) = \frac{4}{95}
\]
Thus, the probability of selecting a yellow marble followed by a purple marble without replacement is \( \frac{4}{95} \).
1. **Determine the total number of marbles**:
There are 20 marbles in total: 4 red, 4 blue, 4 green, 4 yellow, and 4 purple.
2. **Calculate the probability of selecting a yellow marble first**:
The number of yellow marbles is 4, so the probability of selecting a yellow marble is:
\[
P(\text{yellow}) = \frac{\text{Number of yellow marbles}}{\text{Total number of marbles}} = \frac{4}{20} = \frac{1}{5}
\]
3. **Determine the situation after selecting a yellow marble**:
After selecting a yellow marble, the total number of marbles left is 19, and the number of purple marbles remains 4.
4. **Calculate the probability of selecting a purple marble next**:
Now, the probability of selecting a purple marble after having removed one yellow marble is:
\[
P(\text{purple | yellow}) = \frac{\text{Number of purple marbles}}{\text{Total number of remaining marbles}} = \frac{4}{19}
\]
5. **Calculate the total probability of both events occurring**:
The total probability of both selecting a yellow marble and then selecting a purple marble is the product of the two probabilities:
\[
P(\text{yellow then purple}) = P(\text{yellow}) \times P(\text{purple | yellow}) = \left(\frac{1}{5}\right) \times \left(\frac{4}{19}\right) = \frac{4}{95}
\]
Thus, the probability of selecting a yellow marble followed by a purple marble without replacement is \( \frac{4}{95} \).