coexpennan

2021-08-18

An urn contains 3 red and 7 black balls. Players A and B withdraw balls from the urn consecutively until a red ball is selected. Find the probability that A selects the red ball. (A draws the first ball, then B, and so on. There is no replacement of the balls drawn.)

Clelioo

A wins if the first red ball is drawn on the first, third, fifth, or seventh.
We'll count the number of times a red ball appears for the first time. (For example, if a red ball is drawn first, there are (9C2) spots where the other two red balls can be placed. To put it another way, there are (9C2) instances in which A wins on the first draw).
E(1)=(9C2)
E(3)=(7C2)
E(5)=(5C2)
E(7)=(3C2)
When the sum up the number of favorable events and divide by the number of total events.
S=(10C3)
E(x): The number of positive events (position of the first red ball)
S: Total number of occurrences (all possible combinations of the balls)
P(A wins)=$\frac{\left(9C2\right)+\left(7C2\right)+\left(5C2\right)+\left(3C2\right)}{10C3}$
P(A wins)=.05833

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