Bismuth83
A bag contains only red and white balls. The probability of drawing two red balls consecutively is \(\frac{2}{5}\). However, when 6 red balls are replaced with 6 white balls, the probability of drawing two consecutive red balls decreases to \(\frac{3}{20}\).
Select the number of red balls in the bag prior to the replacement under "Red" and the number of white balls in the bag after the replacement under "White". Make only two selections, one from each column.
Let there be r red balls and total t balls.
The probability of drawing two red balls consecutively is \(\frac{2}{5}\).
Thus P = rC2/nC2 = 2/5 ...... r(r-1)/n(n-1) =2/5 ......(i)
However, when 6 red balls are replaced with 6 white balls, the probability of drawing two consecutive red balls decreases to \(\frac{3}{20}\).
Red balls remaining = r-6, while total remain n
Thus P= (r-6)C2/nC2 = (r-6)(r-7)/n(n-1) = 3/20......(ii)
Divide i by ii
r(r-1)/(r-6)(r-7) = 40/15 = 8/3
Substitute the options to see what fits in for r.
r as 16 gives 16*15/10*9 or 8/3
Next substitute r in (i) to get n
16*15/n(n-1) = 2/5
n(n-1) = 8*15*5 = 8*5*3*5 = 5*5*8*3 = 25*24
Hence total =25 and white = 25-16 = 9
After replacing Six red with white, white =9+6 = 15