Question

I have a biased coin that lands heads with probability p and start with an empty urn. I flip the coin n times. Each time the coin lands heads, I add a blue ball to the urn. Each time the coin lands tails, I add a green ball to the urn. After I finish flipping the coin and without knowing the composition of the urn, you draw k balls from the urn one at a time, replacing each ball you draw before drawing another one. If all k of the balls that you draw are blue, what is the probability that all n balls in the urn are blue?

Answer 1

Answer on Question #43478 – Math - Statistics and Probability

I have a biased coin that lands heads with probability $p$ and start with an empty urn. I flip the coin $n$ times. Each time the coin lands heads, I add a blue ball to the urn. Each time the coin lands tails, I add a green ball to the urn. After I finish flipping the coin and without knowing the composition of the urn, you draw $k$ balls from the urn one at a time, replacing each ball you draw before drawing another one. If all $k$ of the balls that you draw are blue, what is the probability that all $n$ balls in the urn are blue?

Remark.

We suppose that we replace each ball with the ball of the same color.

Solution.

The probability that there are $i$ blue ball in the urn equals $p(i) = \binom{n}{i} p^i (1 - p)^{n - i}$ by Bernoulli trail. Suppose that there are $i$ blue balls and $n - i$ green balls. The probability to select one blue ball is $\frac{i}{n}$.

We don't know the composition of the urn, so the probability that the randomly selected ball from the urn is blue equals

$$\begin{array}{l} p_1 = \sum_{i=0}^{n} \frac{i}{n} p(i) = \sum_{i=0}^{n} \frac{i}{n} \binom{n}{i} p^i (1 - p)^{n - i} = \\ = \sum_{i=1}^{n} \binom{n-1}{i-1} p^i (1 - p)^{n - i} = \\ = \sum_{i=0}^{n-1} \binom{n-1}{i} p^{i+1} (1 - p)^{n-1-i} = p. \end{array}$$

Hence the probability that we draw $k$ blue balls is $p_k = p^k$.

This result could also be obtained from geometric distribution.

The probability that all $n$ balls in the urn are blue equals $p_n = p^n$.

If all $k$ of the balls that you draw are blue, then probability that all $n$ balls in the urn are blue equals

$$P = \frac{p_n}{p_k} = p^{n - k}$$

from the formula of conditional probability.

Answer: $P = p^{n - k}$.

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