Color balloons problem
A bag contains some blue and some pink color balloons, all combinations of balloons being equally likely. The total number of color balloons in the bag is 10. If three balloons are drawn at random without replacement and all of them are found to be pink, the probability that the bag contains 1 blue and 9 pink color balloons is
. What is the value of
?
The answer is 69.
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1 solution
let E(i) denotes the event that bag contains i pink and (10-i) blue balloons(i=0,1,2,........10).
Let A denote the event that the three balloons drawn at random from the bag are pink.
we have,
P(Ei) = 1/11(i=0,1,2...10)
P(A/Ei)=0 for i=0,1,2 and P(A/Ei)=iC3/10C3 for i>=3
thus
P(A) = 1/11*1/10C3(3C3 +4C3+...... 10C3)
but
3C3 + 4C3 +5C3.......+10C3= 4C4 + 4C3......10C3
=5C4+5C3+6C3+........+10C3
=11C4
P(A) = (1/11) (1/10C3) 11C4
=1/4.
therefore
P(E9/A)=(P(E9)P(A/E9)) / P(A) =((1/11) *( 9C3/10C3))/(1/4) =14/55
so, a+b=69