Question

In: Statistics and Probability

Consider an urn containing 3 red balls, 5 yellow balls, 7 blue balls, and 6 violet...

Consider an urn containing 3 red balls, 5 yellow balls, 7 blue balls, and 6 violet balls. If 4 balls are drawn at random, what is the probability of drawing one of each color?

Suppose we draw 8 balls from the urn above. What is the conditional probability that 4 of them are blue with respect to the event that none of them are yellow?

Solutions

Expert Solution

Number of red balls in the urn = 3

Number of yellow balls in the urn = 5

Number of blue balls in the urn = 7

Number violet balls in the urn = 6

Total number of balls in the urn = 3+5+7+6 = 21

If 4 balls are drawn at random, probability of drawing one of each color i.e

Probability of drawing one red ball, one yellow ball, one blue ball and one violet ball

= (Number of ways of drawing one red ball from 3 red balls x Number of ways of drawing one yellow ball from 5 yellow balls x Number of ways of drawing one blue ball from 7 blue balls x Number of ways of drawing one violet ball from 6 violet balls) / Number of ways of drawing 4 balls from 21 balls

=

Probability of drawing one red ball, one yellow ball, one blue ball and one violet ball = 0.105263158

If 4 balls are drawn at random, probability of drawing one of each color = 0.105263158

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Suppose we draw 8 balls from the urn above. conditional probability that 4 of them are blue with respect to the event that none of them are yellow

Event A : Drawing 4 blue balls and 4 non blue balls ( 4 of them are blue)

Event B : Drawing non-yellow balls (None of them yellow)

Suppose we draw 8 balls from the urn above. conditional probability that 4 of them are blue with respect to the event that none of them are yellow = P(A|B)

A and B : Event of drawing 4 blue balls and 4 non-blue and non-yellow balls

Number of non-blue and non yellow balls = 3 red balls + 6 violet balls = 9

P(A and B) = Number of ways of drawing 4 blue balls from 7 blue balls x Number of ways of drawing 4 non-blue and non-yellow balls from 9 non-blue and non-yellow balls / Number of ways of drawing 8 balls from 21 balls

P(B) = Number of ways of drawing 8 non-yellow balls from 16 non-yellow balls / Number of ways of drawing 8 balls from 21 balls

Suppose we draw 8 balls from the urn above. conditional probability that 4 of them are blue with respect to the event that none of them are yellow = 0.342657343


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