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3.04 exPECTED VALUE
Objective:
Students determine the expected values of random variables
arising from statistical experiments like the rolling of a
dice, flipping of a coin, spinning a spinner etc.
Recall how we get the expected payoff when we roll a pair
of dice and use this polynomial:
Look at the polynomial for tossing a pair of dice:
(x + x2+ x3 + x4 + x5
+
x6 )2 =
1x2+ 2x3 + 3x4 + 4x5
+
5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + 1X12
If you roll a sum of 3, I gave you 10 dollars, otherwise you
pay me 5 dollars . So, Expected payoff for the sum of 3:
Expected payoff = Expected gain – expected loss
Probability of a sum of 3 = 2/36
Expected gain = 10 x 2/36 = 10 x .05 = .50 cents
Probability for not having a sum of 3 = (36-2)/36 = 34/36
Expected loss = 5 x 34/36 = 5 x .94 = 4.70 dollars
Expected payoff = .50 cents – 4.70 dollars = -4.20 dollars
What about for every sum you roll you win the following and
you loss nothing:
1x2+ 2x3 + 3x4 + 4x5
Roll a sum of
+
Win($)
5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + 1X12
Probability
Expected
payoff
(Win x
probability)
2
3
1
2
1/36=.03
2/36=.05
.03
.10
4
5
6
7
8
9
10
11
12
3
4
5
6
7
8
9
10
11
3/36 =.08
4/36 = .11
5/36 = .14
6/36 = .16
5/36=.14
4/36=.11
3/36=.08
2/36=.05
1/36=.02
.24
.44
.70
.96
.98
.88
.72
.50
.22
Total = $5.77
Expected value winning = total OF THE PROBABILITIES X WIN
SEATWORK:
ROLL THE DICE 5 TIMES. ALL SUMS YOU WIN 5 DOLLARS. BUILD THE
TABLE
3.04a Homework/classwork – expected value
Polynomial for rolling dice 5 times:
Expand ( x + x2 + x3 + X4 + X5 +X6)5 =
65 = 7776 possible outcomes
Roll a sum of
Win($)
2
5
3
5
4
5
6
7
8
9
10
11
12
………
30
Expected value =
5
5
5
5
5
5
Probability
1/7776=
5
5
5
Expected value= Total of expected payoff
Expected
payoff
(Win x
probability)
OTHER EXAMPLES
EXAMPLE 1. Toss five coins. What is the expected value for the number
of heads?
Polynomial : ( H + T)5 = H5 + 5H4T + 10H3T2 +10 H2T3 5HT4 + T5
Total possible outcomes : 25 = 32
Number of heads(H)
Probability
Product:
Number of heads x
probability
0
1/32
0 x 1/32 = 0/32
1
5/32
1 X 5/32 = 5/32
2
10/32
2 x 10/32 = 20/32
3
10/32
3 x 10/32 = 30/32
4
5/32
4X 5/32 = 20/32
5
1/32
5 X 1/32 = 5/32
Total
80/32
Expected value = 0/32 + 5/32 + 20/32 + 30/32 + 20/32 + 5/32
=
0+5+20+30+20+5
32
= 80/32 or 2.5
Interpretation : On the average, you would expect that you
would only get 2.5 heads in flipping 5 coins most of the time
Example 3. The following gives the distribution of lotto betters
who hit the winning numbers in the Mega Millions draw of Dec. 4,
2015. What is the expected number of winning numbers that was
correctly hit during the draw?
Numbers Correctly
Frequency
Product:
6
0
6x0= 0
5
0
5x 0= 0
4
4
4x4 = 16
3
233
3 x233 = 699
2
7756
Matched
2 x 7756 = 15,512
1
161,506
1 x 161,506 = 161506
0
1,234,000
0
Total
1, 403,499
178033
178,033
Expected Value = 1,403,499 = .12
Interpretation: On the average, we expect that only .12 or 0 Winning numbers will be hit.
It’s always a win for the state lottery most of the time in this draw.
Definition – RANDOM VARIABLE
A RANDOM VARIABLE IS ANY OUTCOME THAT IS an INPUT TO get A
NUMBER OR VALUE, LIKE PROBABILITY, number of ways, OR
EXPECTED VALUE. It is usually denoted by a letter like x, y, h,
t, etc.
Input
Outcomes
Rolling a sum of 5 (X)
Getting heads (H)
Getting Tails(T)
a number
To get
Probability of a sum of 5
Number of ways getting
heads
Expected value of tails in 5
tosses
Definition - Expected value
The sum of the outcomes of a random variable multiplied by
its probability.
Tossing a coin to get a head(h) five times, possible outcomes : 25 = 32
Number of heads(h)
Probability (p)
Product:
Number of heads x
probability
hxp
0
1/32
0 x 1/32 = 0/32
1
5/32
1 X 5/32 = 5/32
2
10/32
2 x 10/32 = 20/32
3
10/32
3 x 10/32 = 30/32
4
5/32
4X 5/32 = 20/32
5
1/32
5 X 1/32 = 5/32
Total
80/32
Expected value = 0/32 + 5/32 + 20/32 + 30/32 + 20/32 + 5/32
=
0+5+20+30+20+5
32
= 80/32 or 2.5
Shortcut symbol to get the total or sum :
5
Expected value =
∑ hxp
0
∑
5
or simply
∑hp
0
=80/32
Examples of Random Variables
Example: Tossing a coin: we could get Heads or Tails.
Let's give them the values Heads=0 and Tails=1 and we
have a Random Variable "X":
In short:
X = {0, 1}
Note: We could have chosen Heads=100 and Tails=150 if
we wanted! It is our choice.
So:

We have an experiment (such as tossing a coin)

We give values to each event

The set of values is a Random Variable
Example: Throw a die once
Random Variable X = "The score shown on the top face".
X could be 1, 2, 3, 4, 5 or 6
3.06 Expectation and Variation
Mean
The average of the set of values of a random variable X.
-b
X = ∑𝑋
----N
Where ∑ 𝑋
means sum all the values of x n is the number of values.
Example 1. A random variable X representing the occurrence of
absences:
3, 7, 5, 13, 20, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29
The sum of these numbers is 330
There are fifteen numbers.
The mean is equal to 330 / 15 = 22
The mean of the above numbers is 22