easy Probability
A release-checked easy problem for training Exchangeability, Counting With Indicators, Linearity of Expectation.
A bowl consists of $n$ red and $n$ blue marbles. Marbles are drawn from this bag one-by-one without replacement until there are no marbles left in the bag. The order that the marbles are drawn in is noted. Once the marbles are drawn out, you count the number of runs in your sequence, where a run is defined as a maximal contiguous sequence of the same outcome. For example, $BBRBRR$ has $4$ runs. Find the expected amount of runs in your sequence when $n = 100$.
This Probability problem is tagged Exchangeability, Counting With Indicators, Linearity of Expectation. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.
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