## probability puzzles

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.

Linearity of expectation

50

Each time we draw, there could be a possible color-switch, except the first time we draw, hence 99 possible switches.

Let $X_i$ be an indicator variable, value 1 if $i_{th}$ position has a colour change and zero otherwise, note that $i$ goes from $1$ to $99$.

Total number of switches, $S = X_1 + X_2 + ... + X_{99} = \sum_{i=1}^{99} X_i$

The expected number of color switches is denoted by the sum of these variables, i.e $E[S] = E[X_1 + X_2 + ... + X_{99}]$

Linearity of expectation states that the expected value of the sum of random variables is equal to the sum of their individual expected values.

Hence the answer is $E[S] = E[X_1] + E[X_2] + ... + E[X_{99}]$

Note that each $X_i$ is identical, and the expected value is equal for each $i$, i.e, $E(X_k) = E(X_1)$

Hence $E[S] = E[X_1] + ... + E[X_n] = \sum_{i=1}^{99} E[X_1] = 99 \cdot E[X_1]$

Also, we can calculate the expected value, $E[X_1] = 1 \cdot P(X_1 = 1) + 0 \cdot P(X_1 = 0) = P(X_1 =1)$

Probability of any given color change is (either first is R second is B, or first is B and second is R) $= P(X_1 = 1)= \dfrac{50}{100} \cdot \dfrac{50}{99} + \dfrac{50}{100} \cdot \dfrac{50}{99} = 50/99$

Finally, the answer is $E[S] = \sum_{i=1}^{99} E[X_i] = 99 \cdot E[X_1] = 99 \cdot 50/99 = 50$

**Note**:
$X_i$'s are not independent, but since all the balls are drawn randomly, the probability-distribution is identical for each draw. For instance, if you are given no information about the previous colors, and I ask 'what is the probability that 5th draw has a color change?', your answer will not depend on '5', instead the answer will be same for any index.