Question

five balls are distributed between two urns, labeled A and B. EAch period, one of the five balls is selected at random, and whichever urn it's in, it is moved to the other urn. In the long urn, what fraction of time is urn A empty?

Answer 1

Answer:

Fraction = 0.1667

Step-by-step explanation:

Data:

Let the data be infinite. This means that the situation can go on forever, creating infinite solutions.

Therefore, to make the problem easier, some iterations or recursion are used in the function. Let's see how this is done:

Let, B be the probability that the urn contains i balls. At any point, when the ball is moved from one urn to another, the urn can take 6 values: 0, 1, 2, 3, 4, 5.

Now, as each urn is selected at random, it means that there is an equal chance (50 % or 0.5) value of the urn either to increase or to have the same number of balls throughout.

Thus, the whole situation can be expressed as:

B₀ = 0.5 B₀ + 0.5 B₁

B₁ = 0.5 B₁ + 0.5 B₂

B₂ = 0.5 B₁ + 0.5 B₃

B₃ = 0.5 B₂ + 0.5 B₄

B₄ = 0.5 B₃ + 0.5 B₅

B₅ = 0.5 B₄ + 0.5 B₅

The equivalents will be:

B₅ = B₄

B₃ = B₂

B₂ = B₁

B₁ = B₀

However, the addition can be applied as:

B₀ + B₁ + B₂ + B₃ + B₄ + B₅ + B₆ = 1

This gives B₀ = 0.167 or 1/6