Hard

Probability

You repeatedly draw marbles from a bag containing 50 red and 50 blue marbles until there are no more marbles left, recording the order of red and blue marbles drawn. You then count the number of "runs,'' where a run is defined as any number of consecutive marbles of the same color. For example, $RBBRRRBRR$ contains 5 runs. What is the expected number of runs that you observe?

Notes

Hint