SOLUTION: Two people A and B are to draw alternately one ball at a time from an urn containing 3 white and 2 black balls, drawn balls not being replaced. If A takes the first turn, what is t

Algebra.Com
Question 1183791: Two people A and B are to draw alternately one ball at a time from an urn containing 3 white and 2 black balls, drawn balls not being replaced. If A takes the first turn, what is the probability that A will be the first to draw white?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
i believe the probability that A draws a white ball first is .7.

my thinking:

A gets the white ball on the first draw.
that probability is 3/5 = .6

A gets the white ball on the second draw.
in order for this to happen, both A and B must have drawn a black ball on their first try..
if they do, then there are only white balls left and A is sure to draw a white ball on his second try.
the probability of this happening is 2/5 * 1/4 * 3/3 = 6/60 = .1.

the probability that A is first to get a white ball is therefore .6 + .1 = .7.

since the total probability must be equal to 1, then the probability that B gets the white ball first must be .3.

the ways that this could happen are:

B draws a white ball on the first try.
in order for this to happen, A must have drawn a black ball on the first try.
the probability that B is the first to draw a white ball on the first try is therefore 2/5 * 3/4 = 6/20 = .3

B draws a white ball first on the second try.
in order for this to happen, both A and B must have drawn a black ball on their first try.
sine there are only white balls left, then A is assured to get a white ball on the second try.
the probability that B gets a white ball on the second try is therefore 0.

the probabilities seem to be true.

the probability that A gets a white ball first is .7
the probability that B gets a white ball first is .3
the total probabilities are equal to 1.

this seems reasonable, which is why i think the probability that A gets a white ball first is .7.




Answer by ikleyn(52775)   (Show Source): You can put this solution on YOUR website!
.
Two persons A and B are to draw alternately one ball at a time from an urn containing 3 white and 2 black balls,
drawn balls not being replaced. If A takes the first turn, what is the probability that A will be the first to draw white ?
~~~~~~~~~~~~~~~ 


We may think about the space of events as all possible sequences of the letters W and B of the length 5.


There are 120 permutations of 5 items; but if we consider the distinguishable arrangements of these sequences,

we have only   =  =  = 10 distinguishable orderings, so the space of events has 10 elements.



Next, the "favorable" arrangements are those that EITHER start from W  (and then A takes white ball first),

OR those that start from BBW (and then there is only one such sequence BBWWW, where, again, A takes white ball first).


The number of distinguishable sequences W _ _ _ _  is   =  = 6.

The sequence BBWWW is a unique of that kind.



So, the number of favorable distinguishable sequences is (6+1) = 7, and the total space of events has 10 elements.



THEREFORE, the probability under the problem's question is  P =  =  = 0.7 = 70%.    ANSWER

Solved.

------------------

Good problem (!)

I like it (!)



RELATED QUESTIONS

There are 3 urns containing 2 white and 3 black balls, 3 white and 2 black balls and 4... (answered by richwmiller)
Please help me on this question. Two urns contain white and black balls. Urn 1... (answered by robertb)
Please help me on this question. Thanks! Two urns contain white and black balls. Urn 1... (answered by robertb)
An urn contains 5 white and 7 black balls. Another urn contains 3 white and 9 black... (answered by greenestamps)
Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one... (answered by Fombitz)
You are given two urns, each containing a collection of coloured balls. Urn 1 contains... (answered by ikleyn)
An urn contains 5 white and 7 black balls. Another urn contains 3 white and 9 black... (answered by ikleyn,math_tutor2020)
An urn B1 contains 2 white and 3 black balls and another urn B2 contains 3 white and 4... (answered by greenestamps)
From a bag containing 4 white and 6 black balls, two balls are drawn at random. If the... (answered by math_helper)