A raffle sells 200 tickets at $2, one prize of $300, expected value Step by Step

Step by step

The question

Question: A raffle sells 200 tickets at $2 each.

There is one prize of $300.

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 $300, but we paid $2.

So we are up $298.

The chance is 1/200.

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

The chance is 199/200.

The chances add up to 1: 1/200 and 199/200 make 200/200.

2 Multiply each row

Win: 298 times 1/200 is 298/200.

298 × 1200 = 298200

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 199/200 is minus 398/200.

(−2) × 199200 = −398200

3 Add them

Same bottoms: we add the tops.

A loss takes away.

E = 298 − 398200

Careful!

The chances are not equal.

Each value counts as much as its probability.

Careful: not 298 + (−2)2 = 148 ✗

298 minus 398 is minus 100.

E = −100200

Both 100 and 200 can be divided by 100.

That makes it simpler.

E = −100 ÷ 100200 ÷ 100

So it is minus 1/2.

E = −12

1 divided by 2 is 0.5.

1 ÷ 2 = 0.5

So the expected value is minus 50 cents.

E = −$0.50

Negative: in the long run, you lose 50 cents a ticket.

you lose $0.50 a ticket on average

4 Check

Let's check.

Imagine 200 tickets.

The chances say: 1 win, 199 lose.

in 200 tickets: 1 win, 199 lose

Together: we are down $100.

1 × $298 − 199 × $2 = −$100

Shared over 200 tickets, that is minus 50 cents a ticket.

The same.

So the answer is right!

−$100 ÷ 200 = −$0.50 a ticket

Let's look back.

First: make a table.

Then: multiply each row.

Then: add them.

Last: check.

Answer