Answer :
The probability of picking a red checker, replacing it, and then picking another red checker is [tex]\(\frac{4}{25}\).[/tex]
To solve the problem of picking a red checker from a bag of 9 black checkers and 6 red checkers, replacing it, and then picking another red checker, follow these steps:
The total number of checkers in the bag is:
- [tex]\[9 \text{ (black checkers)} + 6 \text{ (red checkers)} = 15 \text{ checkers}\][/tex]
The probability of picking a red checker on any given draw is:
- [tex]\[\text{Probability of picking a red checker} = \frac{\text{Number of red checkers}}{\text{Total number of checkers}} = \frac{6}{15} = \frac{2}{5}\][/tex]
The probability of picking a red checker on the second draw is the same as the probability of picking a red checker on the first draw, which is [tex]\(\frac{2}{5}\).[/tex]
To find the probability of both events (picking a red checker on the first draw and picking a red checker on the second draw), multiply the probabilities of each independent event:
- [tex]\[\text{Probability of picking a red checker twice} = \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) = \frac{4}{25}\][/tex]
1. Calculate the probability of picking a red checker:
The probability of picking a red checker from the bag is given by the number of red checkers divided by the total number of checkers:
[tex]\[ \text{Probability} = \frac{\text{Number of red checkers}}{\text{Total number of checkers}} = \frac{6}{9 + 6} = \frac{6}{15} = \frac{2}{5} \][/tex]
2. Calculate the probability of picking two red checkers in succession (with replacement):
Since the checker is replaced after each draw, the probability of picking two red checkers in succession is the product of the probabilities of picking a red checker each time:
[tex]\[ \text{Probability of picking two red checkers} = \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) = \frac{4}{25} \][/tex]
Therefore, the probability of picking two red checkers in succession from the bag, with replacement after each pick, is [tex]\( \frac{4}{25} \).[/tex]
