Question

(a) An urn contains$n$white and $m$black balls. The balls are withdrawn one at a time until only those of the same color are left. Show that with probability $n/(n+m)$, they are all white. Hint: Imagine that the experiment continues until all the balls are removed, and consider the last ball withdrawn.

(b) A pond contains$3$distinct species of fish, which we will call the Red, Blue, and Greenfish. There are $r$Red, $b$Blue, and $g$Greenfish. Suppose that the fish are removed from the pond in random order. (That is, each selection is equally likely to be any of the remaining fish.) What is the probability that the Redfish are the first species to become extinct in the pond?

Hint: Write $PR=PRBG+PRGB$, and compute the probabilities on the right by first conditioning on the last species to be removed.

Short answer

a)Use the hint, the wanted event is equivalent to the probability $n/(n+m)$, they are all white.

b)$P\left(R\right)=\frac{bg}{r+b+g}\left(\frac{1}{r+b}+\frac{1}{r+g}\right)$is the probability that the Redfish are the first species to become extinct in the pond

Step-by-step solution

Step 1:Given Information(part a)

An urn contains$n$white and $m$ black balls. The balls are withdrawn one at a time until only those of the same color are left.

Step 2:Explanation(part a)

Follow the hint

By stretching out the perception until the last ball is drawn the probabilities are not changed.

Each stage is similarly reasonable, and each of the $m+n$balls is similarly prone to be the last drawn.

Also, the occasion that the renounces are quick to be drawn (just the white balls remaining) is comparable to the occasion that a white ball is the last drawn

As any of the $m+n$balls is similarly prone to be drawn last, and there are$n$ white balls, the needed probability is

$\frac{n}{n+m}$

Step 3:Final answer(part a)

Use the hint, the wanted event is equivalent to the probability $n/(n+m)$, they are all white.

Step 4:Given Information(part b)

Given that a pond contains $3$distinct species of fish, which we will call the Red, Blue, and Greenfish. There are $r$Red,$b$Blue, and$g$ Greenfish. Suppose that the fish are removed from the pond in random order.

Step 5:Explanation(part b)

Follow the hint

$P\left(R\right)=P\left\{RBG\right\}+P\left\{RGB\right\}$

role="math" localid="1647931375261" $=P\{RB\mid G\text{last}\}P\left(G\text{last}\right)+P\{RG\mid B\text{last}\}P\left(B\text{last}\right)$

Here the principal line is in hint $\left(\right\{RBG\}$ is an occasion that the red, then blue then, at that point, green fish go terminated The subsequent line is valuable to the type of probability of intersections.

The probabilities of conditions are effectively registered as in a), any of the $r+b+g$fish can be the last one got, and the occasion that green or bluefish are the last to go wiped out is comparable to a blue and a green fish be the final remaining one in the lake, thusly:

$P\left(G\text{last}\right)=\frac{g}{r+b+g}P\left(B\text{last}\right)=\frac{b}{r+b+g}$

Step 6:Explanation(part b)

$P\{RB\mid G\text{last}\}$

Every change of $r+b+g$fish can be the request in which the fish are gotten, and from combinatorics, we see that each stage of $r+b$red and bluefish is similarly liable to be the request in which these fish are gotten. all in all, the green fish can be ignored.

Once more, as in a), each of $r+b$fish is similarly liable to be the last drawn, and $RG$is the request iff the last one is blue

$P\{RB\mid G\text{last}\}=\frac{b}{r+b}$

Similarly,

$P\{RG\mid B\text{last}\}=\frac{g}{r+g}$

Step 7:Final Answer(part b)

The probability that the Redfish are the first species to become extinct in the pond is,

$P\left(R\right)=\frac{b}{r+b}\cdot \frac{g}{r+b+g}+\frac{g}{r+g}\cdot \frac{b}{r+b+g}$

$P\left(R\right)=\frac{bg}{r+b+g}\left(\frac{1}{r+b}+\frac{1}{r+g}\right)$