Question
There are 3 green balls, 5 red balls, and 2 blue balls in a box. One ball is drawn at random, replaced, and then another ball is drawn.
What is the probability of drawing a green ball followed by a blue ball?
What is the probability of drawing a green ball followed by a blue ball?
Answers
GPT-4o mini
To find the probability of drawing a green ball followed by a blue ball from the box, we start by determining the total number of balls in the box.
The box contains:
- 3 green balls
- 5 red balls
- 2 blue balls
Adding these together, we have:
\[
3 + 5 + 2 = 10 \text{ balls in total.}
\]
Next, we calculate the probability of drawing a green ball first. The probability \( P(\text{Green}) \) is given by the ratio of the number of green balls to the total number of balls:
\[
P(\text{Green}) = \frac{3}{10}.
\]
Since the ball is replaced before drawing the second ball, the total number of balls remains the same for both draws. Next, we calculate the probability of drawing a blue ball after the green ball has been drawn. The probability \( P(\text{Blue}) \) is given by the ratio of the number of blue balls to the total number of balls:
\[
P(\text{Blue}) = \frac{2}{10} = \frac{1}{5}.
\]
Now we calculate the probability of both events happening in sequence. Since the draws are independent (because of the replacement), we multiply the probabilities of the two events:
\[
P(\text{Green then Blue}) = P(\text{Green}) \times P(\text{Blue}) = \frac{3}{10} \times \frac{1}{5}.
\]
Calculating this gives:
\[
P(\text{Green then Blue}) = \frac{3}{10} \times \frac{1}{5} = \frac{3 \times 1}{10 \times 5} = \frac{3}{50}.
\]
Thus, the probability of drawing a green ball followed by a blue ball is \( \frac{3}{50} \).
The box contains:
- 3 green balls
- 5 red balls
- 2 blue balls
Adding these together, we have:
\[
3 + 5 + 2 = 10 \text{ balls in total.}
\]
Next, we calculate the probability of drawing a green ball first. The probability \( P(\text{Green}) \) is given by the ratio of the number of green balls to the total number of balls:
\[
P(\text{Green}) = \frac{3}{10}.
\]
Since the ball is replaced before drawing the second ball, the total number of balls remains the same for both draws. Next, we calculate the probability of drawing a blue ball after the green ball has been drawn. The probability \( P(\text{Blue}) \) is given by the ratio of the number of blue balls to the total number of balls:
\[
P(\text{Blue}) = \frac{2}{10} = \frac{1}{5}.
\]
Now we calculate the probability of both events happening in sequence. Since the draws are independent (because of the replacement), we multiply the probabilities of the two events:
\[
P(\text{Green then Blue}) = P(\text{Green}) \times P(\text{Blue}) = \frac{3}{10} \times \frac{1}{5}.
\]
Calculating this gives:
\[
P(\text{Green then Blue}) = \frac{3}{10} \times \frac{1}{5} = \frac{3 \times 1}{10 \times 5} = \frac{3}{50}.
\]
Thus, the probability of drawing a green ball followed by a blue ball is \( \frac{3}{50} \).