Hard | Probability |

You are given an urn with 100 balls (50 black and 50 white). You pick balls from urn one by one without replacements until all the balls are out. A black followed by a white or a white followed by a black is "a colour change". Calculate the expected number of colour changes if the balls are being picked randomly from the urn.

Hint

Linearity of expectation

Solution

There are 99 positions. Let X_i be a random variable taking value 1 if i_th position has a colour change and zero otherwise.

We have to find expected value of E[X_1 + X_2 + ... + X_99]

Since all X_i are equivalent, the answer is 99*E[X_i]

E[X_i] = ((50/100)*(50/99)+(50/100)*(50/99)) = 50/99

So, Answer is 50.

We have to find expected value of E[X_1 + X_2 + ... + X_99]

Since all X_i are equivalent, the answer is 99*E[X_i]

E[X_i] = ((50/100)*(50/99)+(50/100)*(50/99)) = 50/99

So, Answer is 50.

Source: Placement test

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