Question

In a box, there are 400 blue marbles and 100 red marbles. When 200 marbles are randomly chosen from the box, let the random variable X represent the number of chosen blue marbles. (a) Find P(X ≥ 102). [You may leave the answer in terms of “combinations” or “factorials”.]

(b) Find P(X = 160) using binomial approximation

Answer 1

Given

Blue marbles = 400

Red marbles = 100

marbles randomly chosen from the box = 200

The probability of selecting a blue marble is

The probability of selecting a red marble is

The random variable X represent the number of chosen blue marbles

a)

By using binomial approximation

= 0.9999 1

And

Using the normal approximation,

The expected number of blue balls out of 200 is = 0.80 * 200 = 160

The variance of the number of blue balls out of 200 is

= 0.80 * 0.20 * 200 = 32

= 0.9999 1

b)

Using binomial approximation, the probability that X will be 160 is

P( X = 160) = 0.0704

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