Expectation & Probability
Measures
When we dealt with statistical samples, we learned to calculate the average (a measure of centrality) and the standard deviation (a measure of dispersal). Here, too, when there are no samples available (i.e., actual results of trials), but only theoretical results of trials (i.e., probability), then we will be able to define and calculate measures.
The measure of centrality in this case is the expectation. The expectation in effect represents the theoretical average, i.e., the average that we expect to obtain if a given trial behaves exactly as we expect according to probability.
Expectation & Probability - Probability Games
You are invited to participate in a game for which the rules are as follows:
- You must throw one of the dice 600 times (every throw is called a “round”).
- In every round, you win a sum of money in dollars equal to the results of the dice toss. The number 1 entitles you to receive $1. The number 2 entitles you to receive $2, and so forth.
- You must pay $1,800 in advance to participate in the game ($3 per round).
Is participating in the game worthwhile?:
The answer is that you cannot know with 100% assurance.
A case is possible where you obtain a few more lower numbers than higher numbers, in which case you will lose money.
On the other hand, if you get higher numbers than lower numbers, you could win a lot of money.
Since you are not given the concrete result of the game before it takes place, you will have to make your decision according to the theoretical result of the game. This is where expectation enters the picture.
The total winning is calculated as follows:
The Winning Number |
Number of Times |
Amount of Winning |
1 |
100 times |
$100 |
2 |
100 times |
$200 |
3 |
100 times |
$300 |
4 |
100 times |
$400 |
5 |
100 times |
$500 |
6 |
100 times |
$600 |
Total |
600 times |
$2,100 |
- The average total winnings per round is $3.50 (i.e., $2,100 divided by 600 ). The conclusion is that it is worthwhile to participate in the game, since you will win $0.50 per round on the average ($3.50 - $3).
- Were a payment of $4 per round required with a total of $2,400, then it would not be worthwhile participating in the game.
- Were a payment of $3.50 per round required, you would be indifferent.
Expected Profit
The theoretical average winnings per round is called the expected profit.
The expected profit in this game is $3.50.
Calculating the Expectation Using a Different Method
Stage 1 – We will calculate the probability of obtaining each number on the dice, and insert it into a table:
Number on the Die
1
2
3
4
5
6
Probability
^{1}/_{6}
^{1}/_{6}
^{1}/_{6}
^{1}/_{6}
^{1}/_{6}
^{1}/_{6}
Stage 2 – Calculating the total winnings and the average winnings per round through the use of probabilities:
The Number Obtained on the Die
The Total Winning in One Round
The Method of Calculation
The Probability of Obtaining the Number
The Number of Rounds in the Game
The Total Winning in the Round
The Total Winning in All Rounds
(1)
(6)
(7)
(8)
(9) = (6) x (7) x (8)
1
^{1}/_{6}
1
$1
$^{1}/_{6}
2
^{1}/_{6}
1
$2
$^{2}/_{6}
3
^{1}/_{6}
1
$3
$^{3}/_{6}
4
^{1}/_{6}
1
$4
$^{4}/_{6}
5
^{1}/_{6}
1
$5
$^{5}/_{6}
6
^{1}/_{6}^{}
1
$6
$1
Total
^{ }
$^{21}/_{6} = $3.50