Two marbles are drawn randomly one after the other without replacement from a jar that contains 8 red marbles, 8 white marbles, and 3 yellow marbles. Find the probability of the following events. Give your answers to at least 4 decimal places, or as an exact expression.

(a) A red marble is drawn first followed by a white marble.

The probability is :

(b) A white marble is drawn first followed by a white marble.

The probability is :

(c) A yellow marble is not drawn at all.

The probability is :

To solve these problems, we need to understand the total number of marbles and how probabilities change depending on whether or not marbles are replaced after being drawn.

### Total Number of Marbles:
The jar contains:
- 8 red marbles
- 8 white marbles
- 3 yellow marbles

The total number of marbles is:
\[
8 + 8 + 3 = 19 \text{ marbles}
\]

### (a) Probability of drawing a red marble first followed by a white marble:

1. **Probability of drawing a red marble first:**
   - There are 8 red marbles.
   - The probability is:
   \[
   \frac{8}{19}
   \]

2. **Probability of drawing a white marble second, after a red marble has been drawn (no replacement):**
   - After one red marble is drawn, 18 marbles remain, including 8 white marbles.
   - The probability is:
   \[
   \frac{8}{18} = \frac{4}{9}
   \]

3. **Combined Probability:**
   - Multiply the probabilities of the two independent events:
   \[
   \frac{8}{19} \times \frac{4}{9} = \frac{32}{171}
   \]

   So, the probability for part (a) is:
   \[
   \frac{32}{171} \approx 0.1871
   \]

### (b) Probability of drawing a white marble first followed by a white marble:

1. **Probability of drawing a white marble first:**
   - There are 8 white marbles.
   - The probability is:
   \[
   \frac{8}{19}
   \]

2. **Probability of drawing another white marble after the first white marble has been drawn (no replacement):**
   - After one white marble is drawn, 18 marbles remain, including 7 white marbles.
   - The probability is:
   \[
   \frac{7}{18}
   \]

3. **Combined Probability:**
   - Multiply the probabilities of the two independent events:
   \[
   \frac{8}{19} \times \frac{7}{18} = \frac{56}{342} = \frac{28}{171}
   \]

   So, the probability for part (b) is:
   \[
   \frac{28}{171} \approx 0.1637
   \]

### (c) Probability that a yellow marble is not drawn at all:

1. **Total Probability without drawing a yellow marble:**
   - There are 16 non-yellow marbles (8 red + 8 white).
   - Probability of drawing the first marble as a non-yellow marble:
   \[
   \frac{16}{19}
   \]
   - Probability of drawing the second marble as a non-yellow marble (after the first non-yellow marble is drawn):
   \[
   \frac{15}{18} = \frac{5}{6}
   \]

2. **Combined Probability:**
   - Multiply the probabilities:
   \[
   \frac{16}{19} \times \frac{5}{6} = \frac{80}{114} = \frac{40}{57}
   \]

   So, the probability for part (c) is:
   \[
   \frac{40}{57} \approx 0.7018
   \]

### Summary of Probabilities:

- **(a)** The probability of drawing a red marble first followed by a white marble is \(\frac{32}{171} \approx 0.1871\).
- **(b)** The probability of drawing a white marble first followed by another white marble is \(\frac{28}{171} \approx 0.1637\).
- **(c)** The probability that a yellow marble is not drawn at all is \(\frac{40}{57} \approx 0.7018\).

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

**Related Questions:**
1. What is the probability of drawing two red marbles in succession?
2. If three marbles were drawn without replacement, what is the probability that all are white?
3. What is the probability of drawing at least one yellow marble in two draws?
4. How would the probabilities change if the marbles were drawn with replacement?
5. What is the probability of drawing one marble of each color in three draws?

**Tip:** When dealing with probabilities, always pay attention to whether the events are independent or dependent, as this affects how you calculate the combined probabilities.

Calculate the probabilities of drawing specific sequences of marbles from a jar containing red, white, and yellow marbles without replacement. Learn the probabilities of drawing a red marble followed by a white marble, two white marbles in succession, and the probability that no yellow marble is drawn at all.

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