Expected Value: Marbles

Calculate expected value for marble scenarios

Calculator Dice Coins Marbles Cards Learn

Expected Red CountCount Prize DrawPrize Sequential ValueSequential

Expected Total from Multiple Draws

red: 3

blue: 2

green: 5

Total: 10 marbles

The Scenario

A bag has 3 red ($5), 2 blue ($3), and 5 green ($1) marbles. You draw 3 marbles without replacement. What is your expected total?

Method 1: Linearity of Expectation

Expected Value of One Draw

Use Linearity for Multiple Draws

By symmetry, each draw has the same expected value:

Answer

E(Total) = $7.80

Method 2: Sum by Color

Alternative approach using expected count of each color:

E(red in 3 draws) = 3 × (3/10) = 0.9 red marbles, worth 0.9 × $5 = $4.50

E(blue in 3 draws) = 3 × (2/10) = 0.6 blue marbles, worth 0.6 × $3 = $1.80

E(green in 3 draws) = 3 × (5/10) = 1.5 green marbles, worth 1.5 × $1 = $1.50

Same answer, different approach.

General Formula

For k draws from a bag with values and counts :

This is just k times the weighted average value of the marbles.

More Examples

Draws	E(red)	E(blue)	E(green)	E(Total)
1	0.3	0.2	0.5	$2.60
2	0.6	0.4	1.0	$5.20
3	0.9	0.6	1.5	$7.80
5	1.5	1.0	2.5	$13.00
10	3.0	2.0	5.0	$26.00

Note: When k = 10 (all marbles), total = 3×$5 + 2×$3 + 5×$1 = $26.00 exactly.

Key Insight

Linearity of expectation makes these problems simple. Whether you calculate per-draw expected value and multiply, or calculate expected counts of each color and sum values, you get the same answer. Choose whichever is easier!