POSC 202A: Lecture 5

Today: Expected Value

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?

Conceptually, draw and label our curve:

-.052

$0

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.

Expected Value

REMEMBER

Expected Value-Is the AVERAGE outcome of a probability distribution. It is our long run expectation of

the expected return of some (social) process.

The outcome in any particular trial, instance, or case, will vary.