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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 it into urn II. Then you randomly select a chip from urn II.

(a) What is the probability that the chip you select from urn II is white?
b) Is selecting a white chip from urn I and selecting a white chip from urn II independent? Justify your answer numerically.

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Answer:

Multiple answers

Step-by-step explanation:

The original urns have:

  1. Urn 1 = 2 red + 4 white = 6 chips
  2. Urn 2 = 3 red + 1 white = 4 chips

We take one chip from the first urn, so we have:

The probability of take a red one is : [tex]\frac{1}{3}[/tex] (2 red from 6 chips(2/6=1/2))

For a white one is: [tex]\frac{2}{3}[/tex](4 white from 6 chips(4/6=(2/3))

Then we put this chip into the second urn:

We have two possible cases:

  • First if the chip we got from the first urn was white. The urn 2 now has 3 red + 2 whites = 5 chips
  • Second if the chip we got from the first urn was red. The urn two now has 4 red + 1 white = 5 chips

If we select a chip from the urn two:

  • In the first case the probability of taking a white one is of:  [tex]\frac{2}{5}[/tex] = 40%  ( 2 whites of 5 chips)
  • In the second case the probability of taking a white one is of:  [tex]\frac{1}{5}[/tex] = 20%  ( 1 whites of 5 chips)

This problem is a dependent event because the final result depends of the first chip we got from the urn 1.

For the fist case we multiply :

[tex]\frac{4}{6}[/tex] x [tex]\frac{2}{5}[/tex] = [tex]\frac{4}{15}[/tex] = 26.66%   ( [tex]\frac{4}{6}[/tex] the probability of taking a white chip from the urn 1, [tex]\frac{2}{5}[/tex]  the probability of taking a white chip from urn two)

For the second case we multiply:

[tex]\frac{1}{3}[/tex] x [tex]\frac{1}{5}[/tex] = [tex]\frac{1}{30}[/tex] = .06%   ( [tex]\frac{1}{3}[/tex] the probability of taking a red chip from the urn 1, [tex]\frac{1}{5}[/tex]   the probability of taking a white chip from the urn two)

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