Suppose you have two urns with poker chips in them. Urn I contains two red chips and four white chips.Urn II contains three red chips and on

Question

Suppose you have two urns with poker chips in them. Urn I contains two red chips and four white chips.Urn II contains three red chips and one white chip. You randomly select one chip from urn I and put itinto urn II. Then you randomly select a chip from urn II.What is the probability that the chip you select from urn II is red?

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Vivian 3 weeks 2022-01-08T09:18:52+00:00 1 Answer 0 views 0

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    2022-01-08T09:20:31+00:00

    Answer:

    P(R_{2}) =\frac{10}{15} = 0.667

    Step-by-step explanation:

    Step 1: Understanding the possible events

    Selecting a chip from Urn I and then adding that chip to Urn II and then selecting a red chip from Urn II can be completed in two ways:

    A. Selecting a red chip from Urn I and adding it to Urn II and then selecting a red chip from Urn II

    B. Selecting a white chip from Urn I and adding it to Urn II and then selecting a red chip from Urn II

    Therefore total probability is:

                                             P(R_{2}) = P(A) + P(B)

    Step 2: Probability of selecting either chip from Urn I

    Urn I contains 2 reds and 4 white chips, that gives a total of 6 chips.

                                                 P(R_{1}) = \frac{2}{6} =\frac{1}{3}

                                                 P(W_{1}) = \frac{4}{6} =\frac{2}{3}

    Step 3: Probability of selecting a red chip from Urn II

    Urn II originally contains 3 reds and 1 white chip, that gives a total of 4 chips.

    Remember: Once a chip is added from Urn I to Urn II the total number of chips will increase in the Urn II

    Case 1: When a red chip is added from Urn I to Urn II

    Red chips    = 4

    White chips = 1

    Total Chips  = 5

                                                      P(R_{2_1}) = \frac{4}{5}

    Case 2: When a white chip is added from Urn I to Urn II

    Red chips    = 3

    White chips = 2

    Total Chips  = 5

                                                      P(R_{2_2}) = \frac{3}{5}

    Therefore the total Probability of selecting a chip from Urn I and then adding that chip to Urn II and then selecting a red chip from Urn II can be calculated as:

                                              P(R_{2}) = P(A) + P(B)

                               P(R_{2}) = P(R_{1}) . P(R_{2_1}) + P(W_{1}) . P(R_{2_2})

                                               P(R_{2}) =\frac{1}{3} . \frac{4}{5}  + \frac{2}{3} .\frac{3}{5}

                                                 P(R_{2}) =\frac{4}{15}  + \frac{2}{5}

                                                P(R_{2}) =\frac{10}{15} = 0.667      

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