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

Step by step

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