4. An urn contains W white balls and B black balls. The balls are withdrawn one

Question

4. An urn contains W white balls and B black balls. The balls are withdrawn one at a time, randomly, until all the balls left in the urn are the same color. What is the probability that the balls left in the urn are white? (This can be viewed as an extinction probability in a simple population model.) Hint: This can be a bit tricky. Consider what happens if you draw all of the balls- how does this relate to drawing until only one color is left? 5. Suppose there are 7 white, 7 red, and 7 black balls in an urn. You randomly draw 5 balls from the urn. What is the probability that you are missing at least one color in your draw? Hint: Try inclusion-exclusion.

Explanation / Answer

question 3

let white=W

red=R

black=B

given- 7W,7R,7B

total ways of selecting 5 balls out of 21=C(21,5)=20349

missing one color means 5 balls are of two colors(either W,R or R,B, or W,B)

probability of missing one color=probability of choosing two color=((C(14,5)+C(14,5)+C(14,5))/C(21,5)

=(2002+2002+2002)/20349=6006/20349=0.295

............................

missing two color means all 5 balls are of same color(either W or ,R or B)

probability of missing two color=probability of choosing one color=(C(7,5)+C(7,5)+C(7,5))/20349=(21+21+21)/20349=63/20349=0.0031

....................

probability of missing atleast two color=probability of missing one color+probability of missing two color

=0.295+0.0031=0.2981

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