Question

An urn holds 60 red marbles and 40 white marbles. Two sets of 30 marbles are drawn with replacement from the urn, and their color is noted. What is the probability that the two sets differ by 8 or more red marbles

Answer 1

Let X and Y denote the number of red marbles obtained in two random
draws of 30 marbles each. Hence
X ~ B(30, 0.6) and Y ~ B(30, 0.6) independently.
Or X ~> N(18, (30 ⋅ 0.6 ⋅ 0.4) ) and Y ~> N(18, (30 ⋅ 0.6 ⋅ 0.4) ).
Therefore, X − Y ~> N(0, 2(30 ⋅ 0.6 ⋅ 0.4) ).

Var=2*30*0.6*0.4=14.4

Sd=(14.4)^(0.5)= 3.79
The required probability is
P( | X − Y | ≥ 8) = P( | X − Y | ≥ 7.5)
≈ 2P(X − Y ≥ 7.5) = 2 Φ(7.5/3.79)
= 2 Φ(1.976) = 2 (0.0241) = 0.0482.

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