Expected Value

1
Expected Value

• Section 3.5

2
Definition

• Lets say that a game gives payoffs
• a1, a2,, an with probabilities
• p1, p2, pn. The expected value ( or
  expectation) E of this game is
• E a1p1 a2p2 anpn.
• Think of expected value as a long term average.

3
American Roulette

• At a roulette table in Las Vegas, you will find
  the following numbers 1 36, 0, 00. There are 38
  total numbers.
• Lets say we play our favorite number, 7.
• We place a 1 chip on 7. If the ball lands in the
  7 slot we win 35 (net winnings). If the ball
  lands on any other number we lose our 1 chip.
• What is the expectation of this bet?
• To answer this question we need to know the
  probability of winning and losing.
• The probability of winning is 1/38. The
  probability of losing is 37/38.
• So the expectation is E 35(1/38)
  (-1)(37/38)
• (35-37)/38 -2/38 -0.053
• What this tells us is that over a long time for
  every 1 we bet we will lose 0.053.
• This is an example of a game with a negative
  expectation. One should not play games when the
  expectation is negative.

4
Example 2

• On the basis of previous experience a librarian
  knows that the number of books checked out by a
  person visiting the library has the following
  probabilities

of books 0 1 2 3 4 5
Probability 0.15 0.35 0.25 0.15 0.05 0.05

• Find the expected number of books checked out
• by a person.
• E 0(0.15)1(0.35)2(0.25)3(0.15)4(0.05)5(0.0
  5)
• E 0 0.35 0.50 0.45 0.20 0.25
• E 1.75

5
Two dice are rolled

• A player gets 5 if the two dice show the same
  number, or if the numbers on the dice are
  different then the player pays 1.
• What is the expected value of this game?
• What is the probability of winning 5? ANSWER
  6/36 1/6.
• What is the probability of paying a 1? ANSWER
  5/6.
• Thus E 5(1/6) (-1)(5/6) 5/6 5/6 0.
• The Expectation is 0. This would be a fair game.

6
Who Wants to be a Millionaire?

• Recall the game show Who Wants to be a
  Millionaire? Hosted by Regis Philbin.
• Lets say youre at the 125,000 question with no
  life-lines. The question you get is the following
• What Philadelphia Eagles head coach has the most
  victories in franchise history?
• A. Earle Greasy Neale
• B. Buddy Ryan
• C. Dick Vermeil
• D. Andy Reid

7
More Millionaire

• If you get the question right you will be at
  125,000.
• If you get the question wrong you fall back to
  32,000.
• Your third option is to walk away with 64,000.
• What to do, what to do?
• Lets do a mathematical analysis.

8
Mathematical analysis

• What is the probability of guessing correctly?
  ANSWER ¼.
• What is the probability of guessing incorrectly?
  ANSWER ¾.
• What is our expectation? ANSWER
• E 125,000(1/4)32,000(3/4) 55,250. This
  is less than the 64,000 walk away value.
• Decision We should walk away.

9
What if we had a 50/50

• Then 2 of the choices would vanish.
• Now the choices left will be A. Greasy Neale and
  D. Andy Reid.
• Now E (1/2)125,000 (1/2)32,000 78,500 gt
  64,000.
• We should give it a shot.
• The answer is A. Greasy Neale (for now, later
  this season Andy Reid will pass Neale).