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Urn Sampling with or without Replacement

blue balls

red balls

drawing type

with replacement

without replacement

show tree

show solution

2 balls are selected at random with replacement from the urn containing 8 blue balls and 6 red balls.Probability of being red for

both: 6 / 14 × 6 / 14 = 36 / 196

one: 6 / 14 × 8 / 14 + 8 / 14 × 6 / 14 = 96 / 196

neither: 8 / 14 × 8 / 14 = 64 / 196

Drawing balls from an urn with or without replacement is a classical problem in probability. Several elementary statistics textbooks use it to introduce the binomial and hypergeometric distributions.

Consider drawing two balls with or without replacement from an urn containing blue and red balls. Let the random variable

represent the number of red balls. Then the possible values of

correspond to the events

{0redballs,1redball,2redballs}

. This Demonstration calculates the probabilities of these three events and shows the probability tree diagram that represents the set of all possible outcomes of the experiment

S={RR,RB,BR,BB}

. The probability of each outcome is shown on its branch.

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