Sure, let's solve these probability problems step-by-step.
First, let's summarize the problem:
- There are 3 red balls and 2 blue balls in a bag.
- The total number of balls is [tex]\(3 + 2 = 5\)[/tex].
- A ball is taken out and then put back (this process is repeated for the second draw).
### (a) Probability that both balls are red
Since the ball is put back after each draw, the probability of drawing a red ball each time remains constant.
1. The probability of drawing the first red ball:
[tex]\[
P(\text{red}) = \frac{3}{5}
\][/tex]
2. The probability of drawing the second red ball:
[tex]\[
P(\text{red}) = \frac{3}{5}
\][/tex]
Since the events are independent:
[tex]\[
P(\text{both red}) = P(\text{first red}) \times P(\text{second red}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}
\][/tex]
### (b) Probability that both balls are the same color
"Same color" means either both are red or both are blue.
1. Probability both balls are red:
[tex]\[
P(\text{both red}) = \frac{9}{25}
\][/tex]
2. Probability both balls are blue:
3
Since the probability of drawing a blue ball remains the same each time:
[tex]\[
P(\text{blue}) = \frac{2}{5}
\][/tex]
[tex]\[
P(\text{both blue}) = P(\text{first blue}) \times P(\text{second blue}) = \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}
\][/tex]
3. Probability both balls are the same color:
[tex]\[
P(\text{both same}) = P(\text{both red}) + P(\text{both blue}) = \frac{9}{25} + \frac{4}{25} = \frac{13}{25}
\][/tex]
### (c) Probability that at least one of the balls is red
The complement of this event is that neither ball is red, which means both balls are blue.
1. Calculate the probability that both balls are blue (already done in part (b)):
[tex]\[
P(\text{both blue}) = \frac{4}{25}
\][/tex]
2. Probability that at least one ball is red is the complement:
[tex]\[
P(\text{at least one red}) = 1 - P(\text{both blue}) = 1 - \frac{4}{25} = \frac{21}{25}
\][/tex]
### Final Answers
(a) The probability that both balls are red is:
[tex]\[
\frac{9}{25}
\][/tex]
(b) The probability that both balls are the same color is:
[tex]\[
\frac{13}{25}
\][/tex]
(c) The probability that at least one of the balls is red is:
[tex]\[
\frac{21}{25}
\][/tex]
