A jar contains m blue balls and n red balls, where m and n are fixed positive in

Question

A jar contains m blue balls and n red balls, where m and n are fixed positive integers. A ball is drawn from the jar randomly (with all possibilities equally likely), and then a second ball is drawn randomly. (a) Describe the sample space of the problem; (b) Is the probability of the second ball being blue lower than () the probability of the first ball being blue? (c) Suppose that there are 16 balls in total, and that the probability that the two balls are the same color is the same as the probability that they are different colors. What are m and n (list all possibilities)?

Explanation / Answer

A) Sample space = {(blue, red) (blue, blue) (red, red) (red, blue)}

B) Probability of first ball being blue = m/(m+n)

Probability of second ball being blue =

(m/(m+n)) x ((m-1)/(m+n-1)) + (n/(m+n))x(m/(m+n-1))

= (mx(m-1) + mn)/((m+n)(m+n-1))

=(m(m+n-1)/((m+n)(m+n-1))

= m/(m+n)

So, the probability of getting a blue ball on first draw is equal to getting a blue ball on second draw

C) Probability of getting same color balls = (mxm+nxn)/((m+n)x(m+n-1)

Probability of getting different colour balls =

((mxn)+(nxm)/(m+n)(m+n-1))

When these two are equal and m+n = 16,

mxm+nxn = mxn+nxm

m2 + n2 -2mn = 0

(m-n)2 = 0

m=n

So, m=8 and n=8

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