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Expected Value
Expected Value-Is the mean outcome of a probability distribution.
It is our long run expectation of the expected return of some (social) process.
Expected Value
The Law of Large Numbers-If a random phenomenon with numerical
outcomes is repeated many times independently, the mean of the actually
observed outcomes approaches the expected value.
Expected Value
To calculate, we need to know:1. The benefit from something occurring (B).2. The probability the benefit occurs (P)3. The cost (benefit) of something not
happening (Bc).4. The probability this cost does not occur (1-P)
Expected Value= (B*P)+ Bc*(1-P)
Expected Value
Expected Value= (B*P)+ Bc*(1-P)
In the overwhelming majority of cases Bc=0.
So, EV reduces to B*P
Expected Value
Expected Value-A random phenomenon that has multiple
outcomes is found by multiplying each outcome by its probability and adding all of
the products.
Expected Value
Expected Value= (B*P)+ Bc*(1-P)
Here we use B to be the net benefit.
In the overwhelming majority of cases Bc=0. Think of this as the return (profit+investment).
So, EV reduces to B*P
Expected Value: Roulette
A roulette wheel has 38 slots, numbered 0,00, and 1-36.
18 are red, 18 are black, and 2 are green.
The wheel is balanced so that the ball is equally likely to land on any slot.
Expected Value: Roulette
Three main bets:One number: win if the number comes up.
One column (or dozen): win if any in the column comes up.
One color: win if the color comes up.
Expected Value: Roulette
The key probabilities are:One number: 1/38One column (or dozen): 12/38Black or Red: 18/38
Expected Value: Roulette
The key bets are:One number: returns $36 (win $35)One column (or dozen): returns $3 (Win $2)Black or Red: returns $2 (win $1)
Expected Value: Roulette
What are the expected values?(Recall, B*P)
One number:
One column:
One color:
Expected Value: Roulette
What are the expected values?Recall, = (B*P)+ Bc*(1-P)
One number: (1/38 * $35)+(37/38*-$1)= (35/38)-(37/38) = -.052
One column: (12/38* $2)+(26/38*$-1)= (24/38)-(26/38) = -.052
One color: (18/38* $1)+(20/38*$-1)= (18/38)-(20/38) = -.052
What does this mean?
Which gives us the best chance of winning money?
Expected Value: Roulette
Shortcut method: return for each $1 bet.Recall, B*P
One number: 1/38 * $36=.947
One column: 12/38* $3= .947
One color: 18/38* $2= .947
What does this mean?
Which gives us the best chance of winning money?
Expected Value: Roulette
Which gives us the best chance of winning money?
To answer this question we can use what we learned about the normal curve to solve for the areas.
How would we do this?
Expected Value: Roulette
How would we do this?
Convert each bet type to standard units and solve for the area that corresponds to a profit.
Expected Value: Roulette
How would we do this?
Next put into Standard Units. Recall
Or 1-.947S.D.
Clearly, we need to find the SD.
Expected Value: Roulette
Clearly, we need to find the SD.
We can use the SD formula from last week.
But, how do we find observations on which to calculate it?
Expected Value: RouletteThe areas (probabilities) from the Z table, differ on each bet:
Red or Black
Column
Number
0.1
.2.3
.4
-20 -10 0 10 20
Recall that underlying distributions converge around the sample mean as the number of trials increase.