Question

You are trying to extract blue marbles that are embedded in a population of red, green, white and black marbles. Blue marbles make up about 10% of the total number. How many marbles must you draw to have:

a. A 50% chance of finding a blue one?

b. A 20% chance of finding two blue ones?

Answer 1

Blue marbles are upto 10% of the total number.

that means if there are n total marbles then 0.1 n are blue marbles.

(a) Let say there are n marbles we draw.

P(finding a blue one atleast) = 0.50

P(failed to find a blue one) = 1 - 0.50 = 0.50

P(there is no blue one) = (1 - 0.10)ⁿ = 0.9ⁿ

0.9ⁿ < 0.5

n ln (0.9) > ln (0.5)

n > 6.6

so atleast 7 should be drawn.

(b) P(Finding two blue ones) = 1 - P(No blue one) - P(one blue one) =1 - ⁿC₀(0.90)ⁿ - ⁿC₁ (0.90)^n-10.1

= 1 - 0.9ⁿ - 0.1n 0.9^n-1

1 - 0.9ⁿ - 0.1n 0.9^n-1 = 0.2

0.9ⁿ + 0.1n 0.9^n-1 = 0.8

0.9^n-1 (0.9 + 0.1 n) = 0.8

so here by trial and error we get

n = 8.32

so here n = 9 there are 20% chance of finding two blue ones.