Question

There are three urns each containing seven red, five green, and three white balls, and two old urns each containing five red, three green, and seven white balls. The urns are identical except for an old or new date stamped beneath the base. If a single red ball is randomly drawn from one of these urns, was it most probably drawn from an old urn or a new urn?

Answer 1

Here there are three urns each containing 7 red, 5 green and 3 white balls.

Two old urns containing 5 red, 3 green and 7 white balls.

Probability of selecting a old urn = 2/5

Probability of selecting a new urn = 3/5

The urns are identical except for an old or new date stamped beneath the base.

If a single red ball is randomly drawn from one of these urns.

Pr(a single red ball is drawn from a old urn) = 5/(5 + 3 + 7) = 1/3

Pr(a single red ball is drawn from a new urn) = 7/(7 + 5 + 3) = 7/15

Probability of selecting a red ball = Pr(a single red ball is drawn from a old urn) * Pr(a old urn is selected) + Pr(a single red ball is drawn from a new urn) * Pr(a new urn is selected)

= 5/15 * 2/5 + 7/15 * 3/5 = 31/75

Pr(the red ball is drawn from old urn) = Pr(a single red ball is drawn from a old urn) * Pr(a old urn is selected) / Probability of selecting a red ball

= (2/5 * 1/3)/(31/75) = (2/15)/(31/75) = 10/31

Pr(The red ball is drawn from new urn) = 1 - 10/31 = 21/31