Expected Value

How to find the expected value of an event: formula, 2 examples, and their solutions.

Formula

Formula

E(X) = x1p1 + x2p2 + x3p3 + ...

E(X): Expected value
x1, x2, x3: Value for each case
p1, p2, p3: Probability for each case

To find the expected value,
multiply the value (xi) and the probability (pi)
for each case,
and add these products.

The expected value E(X) is also called
the mean.

Example 1

Example

Solution

Case 1:
If you get a head,
you get 2 points.

x1 = 2

The probability of getting a head of a coin is
1/2.
So p1 = 1/2.

Then the expected value E(X) is,
x1p1,
2⋅[1/2], ...

Case 2:
If you get a tail,
you lose 1 point.

x2 = -1

The probability of getting a tail of a coin is
1/2.
So p2 = 1/2.

So write, +x2p2,
+(-1)⋅[1/2].

So E(X) = 2⋅[1/2] + (-1)⋅[1/2].

2⋅1 = 2
+(-1)⋅1 = -1

2 - 1 = 1

So E(X) = 1/2.

This means
if a coin is tossed once,
you expect to get 1/2 points.

Example 2

Example

Solution

Case 1: Getting 1 point

x1 = 1

The 1 point area is 3/8 of the whole spinner.
So p1 = 3/8.

So E(X) is,
x1p1,
1⋅[3/8], ...

Case 2: Getting 2 points

x2 = 2

The 2 points area is 3/8 of the whole spinner.
So p2 = 3/8.

So write, +x2p2,
+2⋅[3/8].

Case 3: Getting 3 points

x3 = 3

The 3 points area is 2/8 of the whole spinner.
So p3 = 2/8.

So write, +x3p3,
+3⋅[2/8].

So
E(X) = 1⋅[3/8] + 2⋅[3/8] + 3⋅[2/8].

1⋅3 = 3
+2⋅3 = +6
+3⋅2 = +6

3 + 6 + 6 = 15

So E(X) = 15/8.

This means
if you spin the arrow once,
you expect to get 15/8 points.