(3 points) A box contains 12 red tokens and 11 green tokens. A. If 4 tokens are

Question

(3 points) A box contains 12 red tokens and 11 green tokens. A. If 4 tokens are selected at random, without replacement, determine the probability that at least two of the selected tokens are red. Probability- B. Instead, if 4 tokens are selected at random, with replacement, determine the probability that at least two of the selected tokens are red. Probability C. Instead, if tokens are selected at random, one at a time and with replacement, determine the probability that the first red token appears on draw 11. Probability

Explanation / Answer

1)
12 red and 11 green
total = 23
a)
X = number red ball
P(X>= 2)
= 1 - P(X=0) - P(X = 1)      {hyper geometric distribution }
= 1 - 11C4 / 23C4 - 12C1 * 11C3 / 23C4
= 1 - ((11 * 10 * 9* 8)/24 + 12*11 * 10*9/6) /(23 *22*21*20 /24)
= 0.73913043478

b)
with replacement
X follow binomial distribution
n = 4 , p = 12/23
P(X >= 2)
= 1-P(X = 0) - P(X = 1)
= 1 - (11/23)^4 - 4 * (12/23)* (11/23)^3
= 0.7193799336

c)
first red token on draw 11
geometric distribution
P(X = k) = (1-p)^(k-1) * p
= (11/23)^10*(12/23)
=0.00032666447

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