SOLUTION: There are m white balls and 100 - m black balls in an urn. Two players take turns drawing one ball from the urn until the k-th white ball is drawn. What is the probability that the

Question 1200772: There are m white balls and 100 - m black balls in an urn. Two players take turns drawing one ball from the urn until the k-th white ball is drawn. What is the probability that the first player will draw the k-th white ball and win the game? (m = 68, k = 5)
Answer by asinus(45) (Show Source): You can put this solution on YOUR website!
**1. Define the Event**
* Let's define the event "W" as drawing a white ball.
* Let's define the event "B" as drawing a black ball.
**2. Determine the Initial Probabilities**
* Probability of drawing a white ball on the first draw: P(W) = m / 100 = 68/100 = 0.68
* Probability of drawing a black ball on the first draw: P(B) = (100 - m) / 100 = 32/100 = 0.32
**3. Consider the Winning Scenario**
* The first player wins if they draw the 5th white ball on their turn.
* This implies that in the previous draws, they must have drawn 4 white balls and an appropriate number of black balls.
**4. Calculate the Probability of Winning**
* **Find the probability of drawing 4 white balls and (x - 4) black balls in any order before the 5th white ball (where x is the total number of draws before the 5th white ball):**
* This involves combinations and can be complex to calculate directly.
* **Recognize the Pattern:**
* The probability of drawing the 5th white ball on the player's turn depends on the order in which the previous balls were drawn.
* Since the order doesn't significantly affect the overall probability of the 5th white ball being drawn on the player's turn, we can simplify the calculation.
* **Approximate the Probability:**
* We can approximate the probability of the first player winning by considering the following:
* The probability of drawing a white ball remains relatively constant throughout the game, as the number of white and black balls changes gradually.
* **Simplified Approach:**
* Since the player needs to draw 4 white balls before the 5th white ball, they will have drawn at least 4 other balls (4 white and some black).
* We can assume that the probability of the player drawing the 5th white ball on their turn is approximately 0.68 (the probability of drawing a white ball).
**Therefore, an approximate probability of the first player drawing the 5th white ball and winning the game is 0.68.**
**Important Note:**
* This is an approximation. For a more precise calculation, you would need to consider the changing probabilities of drawing a white ball as the game progresses and use more complex combinatorial calculations.
I hope this explanation helps! Let me know if you'd like to explore a more rigorous calculation.

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