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?

Answer 1

To determine the probability of selecting a yellow marble followed by a purple marble from a jar containing 20 marbles (4 red, 4 blue, 4 green, 4 yellow, and 4 purple), we follow these steps:

1. **Find the probability of selecting a yellow marble first**:
There are 4 yellow marbles out of a total of 20 marbles.
\[
P(\text{Yellow}) = \frac{4}{20} = \frac{1}{5}
\]

2. **Find the probability of selecting a purple marble second**:
After a yellow marble has been selected, there are now a total of 19 marbles remaining (since we are not replacing the yellow marble).
Since we have not removed any purple marbles yet, there are still 4 purple marbles in the jar.
\[
P(\text{Purple | Yellow}) = \frac{4}{19}
\]

3. **Combine the probabilities**:
To find the combined probability of both events occurring (selecting a yellow marble first and then a purple marble), we multiply the probabilities of the two independent events:
\[
P(\text{Yellow and then Purple}) = P(\text{Yellow}) \times P(\text{Purple | Yellow}) = \frac{1}{5} \times \frac{4}{19}
\]

Performing the multiplication:
\[
P(\text{Yellow and then Purple}) = \frac{1 \times 4}{5 \times 19} = \frac{4}{95}
\]

Thus, the probability of selecting a yellow marble first and then a purple marble is \(\boxed{\frac{4}{95}}\).