Expected Value: Marbles

Calculate expected value for marble scenarios

Expected Number of Red Marbles

red: 4

blue: 6

Total: 10 marbles

A bag has 4 red and 6 blue marbles (10 total). If you draw k marbles without replacement, how many red marbles do you expect?

Let = 1 if the i-th draw is red, 0 otherwise.

Total red:

By symmetry, each draw has the same probability of being red:

This works even though draws are dependent (without replacement)!

General Formula

Draw 3 marbles

red

Draw 5 marbles

red

Draw 10 marbles

red

Drawing all 10 gives exactly 4 red (all of them), as expected.

Even though drawing without replacement makes the draws dependent, linearity of expectation still works!

Each position in the draw sequence has the same marginal probability of being red, regardless of what happened in other draws.