Steply

Expected value

Probability · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Make a table
  2. Multiply each row
  3. Add them
  4. Check

The steps for A game: win $10 with probability 1/5, lose $2 otherwise, expected…. Every solution names its steps, so you always know what comes next.

Worked examples

A game: win $10 with probability 1/5, lose $2 otherwise, expected… Easy

The question

Question: In a game, you win $10 with a 1 in 5 chance.

Otherwise you lose $2.

What is the expected value of the game?

1 Make a table

One row for each outcome: its value and its probability.

Win 10, with a chance of 1/5.

Lose: a loss is a negative value, minus 2.

The chance is 4/5.

The chances add up to 1: 1/5 and 4/5 make 5/5.

2 Multiply each row

Win: 10 times 1/5 is 10/5.

10 × 15 = 105

The expected value: each value times its probability, all added up.

It is what we win or lose per game, on average, if we play many times.

E = expected value:
each value × its probability,
all added

Lose: minus 2 times 4/5 is minus 8/5.

(−2) × 45 = −85

3 Add them

Same bottoms: we add the tops.

A loss takes away.

E = 10 − 85

Careful!

The chances are not equal.

Each value counts as much as its probability.

Careful: not 10 + (−2)2 = 4 ✗

10 minus 8 is 2.

E = 25

2 divided by 5 is 0.4.

2 ÷ 5 = 0.4

So the expected value is 40 cents.

E = $0.40

Positive: in the long run, you win 40 cents a game.

you win $0.40 a game on average

4 Check

Let's check.

Imagine 5 games.

The chances say: 1 win, 4 lose.

in 5 games: 1 win, 4 lose

Together: we are up $2.

1 × $10 − 4 × $2 = $2

Shared over 5 games, that is 40 cents a game.

The same.

So the answer is right!

$2 ÷ 5 = $0.40 a game

Let's look back.

First: make a table.

Then: multiply each row.

Then: add them.

Last: check.

Answer
E = $0.40

▶ Watch it solved

Roll a die: win $3 on a 5 or 6, lose $2 otherwise, is it fair Medium

The question

Question: You roll a die.

You win $3 if you roll a 5 or 6.

Otherwise you lose $2.

Is the game fair?

Find the expected value.

1 Make a table

One row for each outcome: its value and its probability.

Win 3, with a chance of 1/3.

Lose: a loss is a negative value, minus 2.

The chance is 2/3.

The chances add up to 1: 1/3 and 2/3 make 3/3.

2 Multiply each row

Win: 3 times 1/3 is 3/3.

3 × 13 = 33

The expected value: each value times its probability, all added up.

It is what we win or lose per game, on average, if we play many times.

E = expected value:
each value × its probability,
all added

Lose: minus 2 times 2/3 is minus 4/3.

(−2) × 23 = −43

3 Add them

Same bottoms: we add the tops.

A loss takes away.

E = 3 − 43

Careful!

The chances are not equal.

Each value counts as much as its probability.

Careful: not 3 + (−2)2 = 0.5 ✗

3 minus 4 is minus 1.

E = −13

1 divided by 3 is 0.333, and more digits.

1 ÷ 3 = 0.33333…

To the nearest cent, the expected value is minus 33 cents.

E ≈ −$0.33

Negative: in the long run, you lose about 33 cents a game.

A fair game has 0.

So this game is not fair.

you lose $0.33 a game on average
not fair

4 Check

Let's check.

Imagine 3 games.

The chances say: 1 win, 2 lose.

in 3 games: 1 win, 2 lose

Together: we are down $1.

1 × $3 − 2 × $2 = −$1

Shared over 3 games, that is about minus 33 cents a game.

The same.

So the answer is right!

−$1 ÷ 3 ≈ −$0.33 a game

Let's look back.

First: make a table.

Then: multiply each row.

Then: add them.

Last: check.

Answer
E ≈ −$0.33

▶ Watch it solved

A raffle sells 100 tickets at $5, one prize of $200, expected value Harder

The question

Question: A raffle sells 100 tickets at $5 each.

There is one prize of $200.

What is the expected value of one ticket?

1 Make a table

One row for each outcome: its value and its probability.

Win: the prize is $200, but we paid $5.

So we are up $195.

The chance is 1/100.

Lose: a loss is a negative value, minus 5.

The chance is 99/100.

The chances add up to 1: 1/100 and 99/100 make 100/100.

2 Multiply each row

Win: 195 times 1/100 is 195/100.

195 × 1100 = 195100

The expected value: each value times its probability, all added up.

It is what we win or lose per game, on average, if we play many times.

E = expected value:
each value × its probability,
all added

Lose: minus 5 times 99/100 is minus 495/100.

(−5) × 99100 = −495100

3 Add them

Same bottoms: we add the tops.

A loss takes away.

E = 195 − 495100

Careful!

The chances are not equal.

Each value counts as much as its probability.

Careful: not 195 + (−5)2 = 95 ✗

195 minus 495 is minus 300.

E = −300100

Both 300 and 100 can be divided by 100.

That makes it simpler.

E = −300 ÷ 100100 ÷ 100

So the expected value is minus $3.

E = −$3

Negative: in the long run, you lose $3 a ticket.

you lose $3 a ticket on average

4 Check

Let's check.

Imagine 100 tickets.

The chances say: 1 win, 99 lose.

in 100 tickets: 1 win, 99 lose

Together: we are down $300.

1 × $195 − 99 × $5 = −$300

Shared over 100 tickets, that is minus $3 a ticket.

The same.

So the answer is right!

−$300 ÷ 100 = −$3 a ticket

Let's look back.

First: make a table.

Then: multiply each row.

Then: add them.

Last: check.

Answer
E = −$3

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. a game: win $10 with probability 1/5, lose $2 otherwise, expected value
    Answer
    E = $0.40
  2. a game: win $6 with probability 1/2, lose $2 otherwise, expected value
    Answer
    E = $2
  3. a game: win $9 with probability 1/3, lose $3 otherwise, expected value
    Answer
    E = $1
  4. roll a die: win $6 on a 6, lose $1 otherwise, expected value
    Answer
    E ≈ $0.17
  5. roll a die: win $12 on a 6, lose $2 otherwise, expected value
    Answer
    E ≈ $0.33
  6. a game: win $8 with probability 1/4, lose $2 otherwise, expected value
    Answer
    E = $0.50
  7. a game: win $4 with probability 1/2, lose $4 otherwise, is it fair
    Answer
    E = $0

Medium

  1. roll a die: win $3 on a 5 or 6, lose $2 otherwise, is it fair
    Answer
    E ≈ −$0.33
  2. roll a die: win $4 on an even number, lose $4 otherwise, is it fair
    Answer
    E = $0
  3. a game: win $8 with probability 1/4, win $4 with probability 1/4, lose $4 otherwise, expected value
    Answer
    E = $1
  4. a game: win $20 with probability 1/10, lose $3 otherwise, expected value
    Answer
    E = −$0.70
  5. pay $2 to play, roll a die, win $9 on a 6, is it fair
    Answer
    E = −$0.50
  6. pay $1 to play, roll a die, win $5 on a 6, is it fair
    Answer
    E ≈ −$0.17
  7. roll a die: win $10 on a 1 or 2, lose $5 otherwise, expected value
    Answer
    E = $0

Harder

  1. a raffle sells 100 tickets at $5, one prize of $200, expected value
    Answer
    E = −$3
  2. a raffle sells 200 tickets at $2, one prize of $300, expected value
    Answer
    E = −$0.50
  3. a raffle sells 50 tickets at $4, one prize of $100, expected value
    Answer
    E = −$2
  4. pay $3 to play, roll a die, win $15 on a 6, is it fair
    Answer
    E = −$0.50
  5. a game: win $5 with probability 1/2, win $1 with probability 1/4, lose $10 otherwise, expected value
    Answer
    E = $0.25
  6. pay $2 to play, roll a die, win $6 on an even number, is it fair
    Answer
    E = $1