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• 7.3 Questions
• 7.4

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More 7.3 questions
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13. Expected Value This is what we were finding
in an earlier example.

• The term for the theoretical winnings per turn.
• Lets denote
• each event as E1, E2,, En
• the probability of each event as
• P1, P2,, Pn
• the payoff of each event as
• X1, X2,, Xn
• The expected value is then
• P1X1 P2 X2 Pn Xn
• A game is only fair if the expected value of the
  game equals the cost of playing the game.

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1. Hack-a-Shaq Source http//www.espn.go.com/nba
/teamstats?teamlal

• Shaquille O'Neal is one of the best players on
  the professional basketball team the Los Angeles
  Lakers.   Shaq, as he is nicknamed, stands 7' 1"
  tall and weighs 330 pounds.  Most of the shots he
  takes are close to the basket, and because he is
  so big other players have a hard time stopping
  him from making baskets.  In fact, he makes 57.2
  of  his shots, which is impressive given that
  most players make about 45.
• In basketball,  when a player trying to make a
  shot is hit on the body by someone on the
  opposing team, thereby causing the player to miss
  the shot, the player gets to take two free shots
  from 15 feet away from the basket.  These shots
  are called foul shots.   Shaq does not shoot foul
  shots very well.  In fact, he makes only 51.3 of
  his foul shots.
• Regular shots are worth two points.  Foul shots
  are worth one point each.
• Because Shaq is less likely to make foul shots,
  one strategy is to foul him whenever he touches
  the ball.  This strategy has been nicknamed the
  "hack-a-Shaq."  Let's see if the hack-a-Shaq pays
  off.

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Problems

• 1.  Calculate the expected value of the number of
  points Shaq scores on one regular shot (not foul
  shots) at the basket (i.e., he makes or misses
  the shot).   Regular shots are the ones he has a
  57.2 chance of making and are worth 2 pts.
• 2.  Assume that all foul shots are independent
  events (examinations of foul shooting records
  suggest this is approximately true).  Calculate
  the expected value of the number of points Shaq
  gets when he shoots two foul shots.
• 3.  Compare the EV for  foul shots to the EV for
  regular shots.   Based on these expected values,
  should opposing teams adopt the hack-a-Shaq?

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2.

• Draw two cards from a deck without replacing
  them. What is the probability of drawing an 8,
  and then a king?

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7.4
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1. How many different ways can 6 people sit in a
row?

• How can we go about solving this problem?

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Write out all of the possibilities

• 123456
• 123465
• 123645
• Etc
• Gets old fast

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Try something simpler What if we have only 3
people

• 123
• 132
• 213
• 231
• 312
• 321

Notice that we have 3 groups of 2 (3)(2)
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How could we do this with 4?
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So we have six options with 1 first. Well have
four groups of 6. (for 1-4) Notice that in this
group of six we have three groups of two. So we
have (4)(3)(2)

• 1234
• 1243
• 1324
• 1342
• 1423
• 1432

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What will happen with 5 people?
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We could also use a tree diagram
Well have 4 groups of 3 groups of 2 (4)(3)(2)
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We can also make connections (This happens to be
my favorite way)

• (4 poss.)(3 poss.)(2 poss.)(1 poss.)
• This is the same as 4!
• n!(n)(n-1)(n-2)(2)(1)

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How many different ways can we sit 6 people in a
row?
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2. Applebees

• Have 6 starters, 6 meals, and 4 desserts. How
  many possible meals could we have?

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3. What happens when we dont use everyone?

• Say there are 6 people in the math club. How many
  different ways can we elect a president and a
  vice president?
• Talk about this with your groups. Any ideas?

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Systematic
We have (6)(5)

• 12 21
• 13 23
• 14 24 etc.
• 15 25
• 16 26

5 ways
6 ways
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Tree
Etc.
Were going to have 6 groups of 5
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Connect to what we did before

• (6 poss.)(5 poss.)

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In your groups determine how many ways we could
have a Pres, VP and Treasurer
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4. Permutations

• If we remember work from College Algebra
• (6)(5)(4) (654321)/(321)
• (6!)/(3!)
• (6!)/((6-3)!)

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• We refer to the number of different subsets of
  size r that we can make from a set of size n as
  the number of permutations of n things taken r at
  a time.
• The shorthand is nPr

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