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 selecting a purple marble
Answers
GPT-4o mini
To find the probability of selecting a yellow marble followed by a purple marble without replacement, we can break the problem down into two parts.
1. **Selecting a yellow marble:**
There are 4 yellow marbles out of a total of 20 marbles. So, the probability of selecting a yellow marble on the first draw is:
\[
P(\text{Yellow}) = \frac{4}{20} = \frac{1}{5}
\]
2. **Selecting a purple marble after selecting a yellow marble:**
After one yellow marble has been selected, there will be 19 marbles left in the jar. Since we have not removed any purple marbles yet, the number of purple marbles remains 4. The probability of selecting a purple marble now is:
\[
P(\text{Purple | Yellow}) = \frac{4}{19}
\]
3. **Calculating the combined probability:**
The combined probability of both events happening (selecting a yellow marble first and then selecting a purple marble) is found by multiplying the probabilities of the two independent events:
\[
P(\text{Yellow and Purple}) = P(\text{Yellow}) \times P(\text{Purple | Yellow})
\]
Plugging in the values we calculated:
\[
P(\text{Yellow and Purple}) = \frac{1}{5} \times \frac{4}{19} = \frac{4}{95}
\]
Thus, the probability of selecting a yellow marble followed by a purple marble without replacement is:
\[
\frac{4}{95}
\]
1. **Selecting a yellow marble:**
There are 4 yellow marbles out of a total of 20 marbles. So, the probability of selecting a yellow marble on the first draw is:
\[
P(\text{Yellow}) = \frac{4}{20} = \frac{1}{5}
\]
2. **Selecting a purple marble after selecting a yellow marble:**
After one yellow marble has been selected, there will be 19 marbles left in the jar. Since we have not removed any purple marbles yet, the number of purple marbles remains 4. The probability of selecting a purple marble now is:
\[
P(\text{Purple | Yellow}) = \frac{4}{19}
\]
3. **Calculating the combined probability:**
The combined probability of both events happening (selecting a yellow marble first and then selecting a purple marble) is found by multiplying the probabilities of the two independent events:
\[
P(\text{Yellow and Purple}) = P(\text{Yellow}) \times P(\text{Purple | Yellow})
\]
Plugging in the values we calculated:
\[
P(\text{Yellow and Purple}) = \frac{1}{5} \times \frac{4}{19} = \frac{4}{95}
\]
Thus, the probability of selecting a yellow marble followed by a purple marble without replacement is:
\[
\frac{4}{95}
\]