Probability ( Fundamental )

假設現在有27個藍色波，37個黃色波，36個紅色波，n=100。現在random地抽 5個波 at a time with replacement.


= = = = =
(1)
B = blue ball
N = not blue ball

由於有 replacement，每次都係由 100 balls 抽出 1 ball。每一次：
P(B) = 27/100
P(N) = 1 - (27/100) = 73/100

P(at most 2 B)
= P(0B5N) + P(1B4N) + P(2B3N)
= (73/100)^5 + 5C1*(27/100)*(73/100)^4 + 5C2*(27/100)^2*(73/100)^3
= 4371384029/5000000000 or 0.8742768058


= = = = =
(2)
P([at most 2B] and [at least 2N])
= P([at most 2B] and [at most 3B])
= P(at most 2B)
= 0.8742768058

P(at least 2N)
= P(0B5N) + P(1B4N) + P(2B3N) + P(3B2N)
= 0.8742768058 + 5C3*(27/100)^3*(73/100)^2
= 0.9791675128

P(at most 2B | at least 2N)
= P([at most 2B] and [at least 2N]) / P(at least 2N)
= 0.8742768058/0.9791675128
= 820301/918716 or 0.8928776684

to樓上: 一次抽5個

係一次抽 5 個定係每次抽 1 個兼每次 replacement?