# Calculating Probability for a Game of Picking White Balls: Three Players

• stephenranger
In summary: You seem to have a mistake there.Oh. I'm sorry. It should be P_A=\sum\limits_{n=1}^{\infty}(4/5)^{3n-1}(1/5) = 1/5\sum\limits_{n=1}^{\infty}(4/5)^{3n}=\frac{1}{5}\left(\frac{1}{1-64/125}\right)=\frac{125}{189}\approx0.661375In summary, the probability that the game ends before any of the players has picked twice is 61/125 = 0.488, and the probability for each player to win the game is approximately 0
stephenranger

## Homework Statement

Three friends play a game in which one picks blind–folded from a bag containing white and
black balls. In the bag there are four black balls and one white ball. The player
whose turn it is picks one ball. If the ball is white the player has won; otherwise
the ball is returned to the bag and the next player gets the turn. The turn rotates
until the white ball is picked.
a) What is the probability that the game ends before any of the players has
picked twice?
b) Let the players be A, B, and C, in this order. What is each player’s probability
of winning the game?

## The Attempt at a Solution

The following is my solution. But I'm not sure if it is correct. Please correct me if there's any mistake. Thanks.

a)
The probability of the 1st player picks the white ball is P1 = 1/5
The probability of the 1st player picks a black ball and then the 2nd player picks the white ball is P2 = (4/5)x(1/5)
The probability of the 1st player picks a black ball and then the 2nd player picks a black ball and then the 3rd picks the white ball is P3 = (4/5)x(4/5)x(1/5)

So the probability that the game ends before any of the players has picked twice is: P = P1+P2+P3 = (1/5) + (4/5)x(1/5) + (4/5)x(4/5)x(1/5) = 61/125 = 0.488

b)
The probability that A picks the white ball is PA = 1/5
The probability that B picks the white ball is PB = (4/5)x(1/5)
The probability that C picks the white ball is PC = (4/5)x(4/5)x(1/5)

[/B]

I don't think your answer is correct for part b). You seem to have tried to calculate the individual probabilities of each player winning after one pick only. However, can't the game go on for multiple rounds before a winner happens?

Last edited:
Instead of duplicating a thread, you can always ask a moderator to move it to a more appropriate forum if needed.

stephenranger said:
Oh yeah. I'm sorry because posting in the precalculus mathematics, no one gives me a decent answer and otherwise this problem is supposed to be beyond pre-calculus mathematics, right?
You could try responding to my post there.

Just calculate the probability of the player A winning on the n-th round $P_A(n)$. Then the probability for A to win the game is $P_A=\sum\limits_{n=1}^{\infty}P_A(n)$

Last edited:
haruspex said:
You could try responding to my post there.
Ok

Delta² said:
Just calculate the probability of the player A winning on the n-th round $P_A(n)$. Then the probability for A to win the game is $P_A=\sum\limits_{n=1}^{\infty}P_A(n)$
So, the probability that A picks the white ball at:
the 1st round: 1/5
the 2nd round: (4/5)3.(1/5)
the 3rd round: (4/5)6.(1/5)
the 4th round: (4/5)9.(1/5)
............
............
the n-th round: (4/5)3n.(1/5)

Therefore: ∑(4/5)3n.(1/5) when n runs from 0 → ∞ ≈ 0.32

stephenranger said:
Therefore: ∑(4/5)3n.(1/5) when n runs from 0 → ∞ ≈ 0.32
≈0.4

## 1. What is probability?

Probability is a measure of the likelihood of an event occurring. It is usually expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

## 2. How do you calculate probability?

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This can be represented as P(event) = number of favorable outcomes / total number of possible outcomes.

## 3. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and predictions, while experimental probability is based on actual outcomes from experiments or observations.

## 4. What is the difference between independent and dependent events?

Independent events are events that do not affect each other's probability, while dependent events are events that do affect each other's probability. In other words, the outcome of one event does not impact the outcome of the other in independent events, but it does in dependent events.

## 5. How can probability be used in real life situations?

Probability can be used in many real-life situations, such as predicting the likelihood of winning a game, making business decisions, and determining risk in insurance or financial investments. It can also be used to analyze data and make predictions in many fields, including science, economics, and medicine.

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