Round all answers to 4 decimal places.

a.  A bag contains 8 𝑤ℎ𝑖𝑡𝑒 marbles, 10 𝑦𝑒𝑙𝑙𝑜𝑤 marbles, and 7 𝑏𝑙𝑢𝑒 marbles. If a marble is drawn from the bag, replaced, and another marble is drawn, what is the probability of drawing first a 𝑤ℎ𝑖𝑡𝑒 marble and then a 𝑏𝑙𝑢𝑒 marble?

b.  A bag contains 10 𝑟𝑒𝑑 marbles, 7 𝑏𝑙𝑢𝑒 marbles, and 8 𝑦𝑒𝑙𝑙𝑜𝑤 marbles. If two different marbles are drawn from the bag , what is the probability of drawing first a 𝑟𝑒𝑑 marble and then a 𝑦𝑒𝑙𝑙𝑜𝑤 marble?

### a. Probability of drawing first a white marble and then a blue marble (with replacement):

1. The total number of marbles in the bag is:
\[
8 \, (\text{white}) + 10 \, (\text{yellow}) + 7 \, (\text{blue}) = 25 \, (\text{total marbles})
\]

2. The probability of drawing a white marble first is:
\[
P(\text{white}) = \frac{8}{25}
\]

3. Since the marble is replaced, the total number of marbles remains 25. The probability of drawing a blue marble second is:
\[
P(\text{blue}) = \frac{7}{25}
\]

4. The combined probability of drawing a white marble first and a blue marble second (with replacement) is the product of the two individual probabilities:
\[
P(\text{white and blue}) = P(\text{white}) \times P(\text{blue}) = \frac{8}{25} \times \frac{7}{25} = \frac{56}{625}
\]

5. Rounded to 4 decimal places:
\[
P(\text{white and blue}) \approx 0.0896
\]

---

### b. Probability of drawing first a red marble and then a yellow marble (without replacement):

1. The total number of marbles in the bag is:
\[
10 \, (\text{red}) + 7 \, (\text{blue}) + 8 \, (\text{yellow}) = 25 \, (\text{total marbles})
\]

2. The probability of drawing a red marble first is:
\[
P(\text{red}) = \frac{10}{25}
\]

3. Since the marble is not replaced, the total number of marbles left is now 24. The probability of drawing a yellow marble second is:
\[
P(\text{yellow after red}) = \frac{8}{24} = \frac{1}{3}
\]

4. The combined probability of drawing a red marble first and a yellow marble second (without replacement) is:
\[
P(\text{red and yellow}) = P(\text{red}) \times P(\text{yellow after red}) = \frac{10}{25} \times \frac{1}{3} = \frac{10}{75} = \frac{2}{15}
\]

5. Rounded to 4 decimal places:
\[
P(\text{red and yellow}) \approx 0.1333
\]

---

Would you like further details or have any questions about these calculations?

Here are five related questions to expand on this:

1. What is the probability of drawing two white marbles consecutively with replacement?
2. What is the probability of drawing a yellow marble first and a blue marble second without replacement?
3. What would be the probability of drawing a red marble followed by a blue marble without replacement?
4. How does the probability change if three marbles are drawn in succession without replacement?
5. What is the probability of drawing two yellow marbles consecutively without replacement?

**Tip:** For probabilities involving independent events (like drawing with replacement), the total number of outcomes remains the same after each draw. For dependent events (like drawing without replacement), the total number of outcomes changes after each draw.

A bag contains 8 white marbles, 10 yellow marbles, and 7 blue marbles. If a marble is drawn from the bag, replaced, and another marble is drawn, what is the probability of drawing first a white marble and then a blue marble? A bag contains 10 red marbles, 7 blue marbles, and 8 yellow marbles. If two different marbles are drawn from the bag, what is the probability of drawing first a red marble and then a yellow marble?

Learn how to calculate the probability of drawing specific marbles from a bag, with or without replacement. This detailed solution covers the probability of drawing first a white marble and then a blue marble with replacement, as well as the probability of drawing first a red marble and then a yellow marble without replacement. Suitable for students in Grades 9-12.

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