Pay $1 to play, roll a die, win $5 on a 6, is it fair Step by Step

Step by step

The question

Question: It costs $1 to play.

You roll a die and win $5 if you roll a 6.

Is the game fair?

Find the expected value.

1 Make a table

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

Win: the prize is $5, but we paid $1.

So we are up $4.

The chance is 1/6.

Lose: a loss is a negative value, minus 1.

The chance is 5/6.

The chances add up to 1: 1/6 and 5/6 make 6/6.

2 Multiply each row

Win: 4 times 1/6 is 4/6.

4 × 16 = 46

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 1 times 5/6 is minus 5/6.

(−1) × 56 = −56

3 Add them

Same bottoms: we add the tops.

A loss takes away.

E = 4 − 56

Careful!

The chances are not equal.

Each value counts as much as its probability.

Careful: not 4 + (−1)2 = 1.5 ✗

4 minus 5 is minus 1.

E = −16

1 divided by 6 is 0.167, and more digits.

1 ÷ 6 = 0.16667…

To the nearest cent, the expected value is minus 17 cents.

E ≈ −$0.17

Negative: in the long run, you lose about 17 cents a game.

A fair game has 0.

So this game is not fair.

you lose $0.17 a game on average

not fair

4 Check

Let's check.

Imagine 6 games.

The chances say: 1 win, 5 lose.

in 6 games: 1 win, 5 lose

Together: we are down $1.

1 × $4 − 5 × $1 = −$1

Shared over 6 games, that is about minus 17 cents a game.

The same.

So the answer is right!

−$1 ÷ 6 ≈ −$0.17 a game

Let's look back.

First: make a table.

Then: multiply each row.

Then: add them.

Last: check.

Answer