A bag contains 1 red ball. 1 white ball and 5 black balls, Matthew and Lisa take turns to draw a randomly from the bag with replacement, the one who first draws a red or white ball wins the game game starts with Matthew. Find the probability that Matthew wins the game by drawing a red ball.

To determine the probability that Matthew wins the game by drawing a red ball, we can use the concept of geometric probability. Let's break down the problem step by step.

1. **Determine the probabilities for each type of ball:**
   - There is 1 red ball, 1 white ball, and 5 black balls.
   - The total number of balls is \(1 + 1 + 5 = 7\).

   Therefore, the probability \(P(\text{red})\) of drawing a red ball is:
   \[
   P(\text{red}) = \frac{1}{7}
   \]

   The probability \(P(\text{white})\) of drawing a white ball is:
   \[
   P(\text{white}) = \frac{1}{7}
   \]

   The probability \(P(\text{black})\) of drawing a black ball is:
   \[
   P(\text{black}) = \frac{5}{7}
   \]

2. **Calculate the probability that Matthew wins:**

   Matthew wins immediately if he draws a red ball on his first turn. This probability is:
   \[
   P(\text{Matthew wins on 1st turn}) = P(\text{red}) = \frac{1}{7}
   \]

3. **Consider subsequent rounds:**

   If Matthew draws a black ball (probability \( \frac{5}{7} \)), the game continues to Lisa. Lisa will also draw a ball with the same probabilities. For Matthew to win, Lisa must not draw a red or white ball (i.e., she must draw a black ball, probability \( \frac{5}{7} \)) and the game will return to Matthew. This cycle repeats.

4. **Geometric series for repeated attempts:**

   The probability that both draw a black ball in one full cycle (one turn each for Matthew and Lisa) is:
   \[
   P(\text{both black}) = \left(\frac{5}{7}\right)^2 = \frac{25}{49}
   \]

   Therefore, the probability that the first round results in Matthew drawing a black ball and Lisa drawing a black ball and the game returning to Matthew is \( \frac{25}{49} \).

   The game resets to the original state after each cycle where both draw black balls, hence we can consider it as a geometric series where the first term is the probability that Matthew wins on his first turn, and the common ratio is the probability that the game resets.

   The total probability that Matthew wins is then:
   \[
   P(\text{Matthew wins}) = P(\text{Matthew wins on 1st turn}) + P(\text{both draw black}) \times P(\text{Matthew wins})
   \]
   \[
   P(\text{Matthew wins}) = \frac{1}{7} + \frac{25}{49} \times P(\text{Matthew wins})
   \]

   Let \( x = P(\text{Matthew wins}) \), then:
   \[
   x = \frac{1}{7} + \frac{25}{49} x
   \]
   \[
   x - \frac{25}{49} x = \frac{1}{7}
   \]
   \[
   \left(1 - \frac{25}{49}\right) x = \frac{1}{7}
   \]
   \[
   \frac{24}{49} x = \frac{1}{7}
   \]
   \[
   x = \frac{1}{7} \times \frac{49}{24}
   \]
   \[
   x = \frac{49}{168}
   \]
   \[
   x = \frac{7}{24}
   \]

Therefore, the probability that Matthew wins the game by drawing a red ball is \(\frac{7}{24}\).

Would you like more details or have any questions?

Here are 8 related questions to further your understanding:

1. What is the probability that Lisa wins the game by drawing a white ball?
2. How does the probability change if there are no black balls?
3. What if the number of black balls is increased to 7? What would be the new probability of Matthew winning?
4. How would you calculate the expected number of turns for the game to end?
5. What is the probability that the game lasts exactly 3 turns?
6. How would the problem change if balls are drawn without replacement?
7. What if the game starts with Lisa instead of Matthew?
8. How would the strategy change if the winning ball had to be a specific color, such as blue?

**Tip:** When solving probability problems involving cycles or repeated trials, setting up equations with geometric series can greatly simplify the calculation process.

Explore the probability problem where Matthew and Lisa draw balls from a bag with replacement. Learn how to calculate the probability that Matthew wins the game by drawing a red ball, including step-by-step solutions and geometric series analysis. Suitable for Grades 10-12.

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