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Probability for Computer Scientists
CS109
Cynthia Lee
Socrative classroom:
2
Today’s Topics
 Last week:
› Conditional probability, independence
› Conditional independence
› Odds
› Random variables, Expectation, PMF and CDF
› (wow, busy week!!)
TODAY: MORE EXPECTATION AND MORE DISCRETE RVS!
› Expectation example
› Random Variables
• Bernoulli RV
• What is variance? Standard deviation?
• Binomial RV
 Rest of the week:
› More random variables: discrete and then continuous
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Casino Examples
EXPECTATION
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Expectation example: Coin flipping
 In this game, you have a fair coin, and you keep flipping it until you get
tails. Let n = the number of heads flipped before we get tails. Your
winnings are then computed as 2n.
 How much would it make sense to pay to play this game?
A. < $1
B. $1 - $10
C. $10 - $100
D. > $100
E. I would not agree to play this game.
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Expectation example: Coin flipping
 In this game, you have a fair coin, and you keep flipping it until you get
tails. Let n = the number of heads flipped before we get tails. Your
winnings are then computed as 2n.
 How much would it make sense to pay to play this game?
› Make a random variable X = your winnings
› We want to find E[X], or how much we “expect” to win when we play
the game.
• It would make sense to pay anything up to the expected value for
the game.
• It would not make sense to pay any more than the expected value
for the game (though that rules out nearly all casino games and
state lotteries).
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Expectation example: Coin flipping
 In this game, you have a fair coin, and you keep flipping it until you get
tails. Let n = the number of heads flipped before we get tails. Your
winnings are then computed as 2n.
 How much would it make sense to pay to play this game?
› Make a random variable X = your winnings
› We want to find E[X]:
 According to expectation, you should be willing to pay any amount to
play this game.
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Expectation example: Roulette
 In roulette, p(red) = 18/38 (18 red slots, 18 black slots, and 2 green).
 Your strategy is as follows:
1. Your bet Y starts at Y = $1
2. Bet Y.
3. If you win, STOP.
4. If you lose, Y *= 2 (double down), goto 2.
 Does it make sense to pay to play this game?
A. Yes
B. No
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Expectation example: Roulette
 In roulette, p(red) = 18/38 (18 red slots, 18 black slots, and 2 green).
 Your strategy is as follows:
1. Your bet Y starts at Y = $1
2. Bet Y.
3. If you win, STOP.
4. If you lose, Y *= 2 (double down), goto 2.
 Does it make sense to pay to play this game?
› Make a random variable Z = your winnings (for the whole strategy, so
less earlier losses)
› We want to find E[Z]:
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Expectation example: Roulette
 In roulette, p(red) = 18/38 (18 red slots, 18 black slots, and 2 green).
 Your strategy is as follows:
1. Your bet Y starts at Y = $1
2. Bet Y.
3. If you win, STOP.
4. If you lose, Y *= 2 (double down), goto 2.
 Does it make sense to pay to play this game?
› Make a random variable Z = your winnings (for the whole strategy, so
less earlier losses)
› We want to find E[Z]:
Now you have E[Z], interpret it: does it
make sense to pay to play this game?
A. Yes
B. No
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Expectation example: Roulette
 In roulette, p(red) = 18/38 (18 red slots, 18 black slots, and 2 green).
 Your strategy is as follows:
1. Your bet Y starts at Y = $1
2. Bet Y.
3. If you win, STOP.
4. If you lose, Y *= 2 (double down), goto 2.
 Does it make sense to pay to play this game?
› Make a random variable Z = your winnings (for the whole strategy, so
less earlier losses)
› We want to find E[Z]:
Yes. You win little each time, but could
play again and again and just keep
accruing money.
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Expectation example: Roulette
 In roulette, p(red) = 18/38 (18 red slots, 18 black slots, and 2 green).
 Your strategy is as follows:
1. Your bet Y starts at Y = $1
2. Bet Y.
3. If you win, STOP.
4. If you lose, Y *= 2 (double down), goto 2.
 Does it make sense to pay to play this game?
› Make a random variable Z = your winnings (for the whole strategy, so
less earlier losses)
› We want to find E[Z]:
Or No, if the opportunity cost is too
high (you could make more money per
hour doing something else).
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Expectation example: Roulette
 In roulette, p(red) = 18/38 (18 red slots, 18 black slots, and 2 green).
 Your strategy is as follows:
1. Your bet Y starts at Y = $1
2. Bet Y.
3. If you win, STOP.
4. If you lose, Y *= 2 (double down), goto 2.
 So why doesn’t everyone just play this strategy in roulette all the
time?
› You can’t—you have limited amounts of money to keep doubling
down. It only works if you can keep doubling down without limit.
› You can’t—casinos have done this same math and set maximum
bet amounts in part to prevent this strategy.
› They can also just kick you out any time they don’t like what you’re
doing.
Variance
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Variance
 Here are the PMFs for three different games
› x axis is RV X = winnings
› y axis is p(X=x)
 Which game would you play?
A. First one because it has the greatest chance of maximum $ win
B. Last one because it has the least chance of minimum $ win
C. Doesn’t matter, they all have same E[X]
D. Other/none/more than one
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Variance
 DEFINITION: VARIANCE
Variance Example: ship sizes
 Here are several observed ship sizes:
› 6.3m, 9.6m, 26.7m, 37m, 12.5m, 24m, 40m, 20.5m
› We’ll line them up and rotate for easier comparison
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Variance Example: ship sizes
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 Here are several observed ship sizes:
› 6.3m, 9.6m, 26.7m, 37m, 12.5m, 24m, 40m, 20.5m
› Average = 22.075m
› Now we want to calculate the square of how far from average each ship is:
• (6.3 - 22.075)2
• (40.0 - 22.075)2
• …
› Take the average of the above….and that’s variance
• “How spread out (squared), on average?”
Variance:
full derivation of a more convenient formula
Properties of Variance:
not quite as nice as linearity of Expectation, but something…
Variance Example: ship sizes
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 We add a U-Haul trailer to all the ships:
› +14.5m added to each: 6.3m, 9.6m, 26.7m, 37m, 12.5m, 24m, 40m, 20.5m
› Average changes: 22.075m  36.575m
› Now how much does the variance change?
Variance Example: ship sizes
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 We add a U-Haul trailer to all the ships:
› +14.5m added to each: 6.3m, 9.6m, 26.7m, 37m, 12.5m, 24m, 40m, 20.5m
› Average changes: 22.075m  36.575m
› Now how much does the variance change?
• VARIANCE DOES NOT CHANGE
Variance Example: ship sizes
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 We add a U-Haul trailer to all the ships:
› +14.5m added to each: 6.3m, 9.6m, 26.7m, 37m, 12.5m, 24m, 40m, 20.5m
› Average changes: 22.075m  36.575m
› Now how much does the variance change?
• VARIANCE DOES NOT CHANGE
Variance Example: ship sizes
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 Now let’s say we want to make toy models of these ships. Each model will
be 1/10th the actual size.
› Actual sizes: 6.3m, 9.6m, 26.7m, 37m, 12.5m, 24m, 40m, 20.5m
› New average = 22.075m / 10 = 2.2075m
 The new variance is ____________ the old variance.
A. …more than 10x larger than…
B. …10x larger than…
C. …equal to…
D. …1/10th of…
E. …less than 1/10th of…
Variance example: dice
Standard Deviation:
a close cousin of variance
Bernoulli Trials
BERNOULLI
RANDOM VARIABLES
Your Random Variable Quick Guide!
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Bernoulli Random Variables
 In terms of p, what is E[X] for any Bernoulli random variable?
A. p
B. (1-p)
C. p2
D. Other/none/more than one
Bernoulli Random Variables
Binomial Random Variables
Birds of a Feather
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Giraffe
Herd
Bird
Flock
Bernoulli RV
Binomial RV
Your RV Quick Reference Guide
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Bernoulli RV
If the possible values for an RV X are
either 0 or 1, then X is a Bernoulli RV.
(It’s black & white, because it’s
boolean!)
Binomial RV
Is an RV that describes the behavior
of n trials of a Bernoulli RV with
probability p.
(It’s a whole herd/flock/pride of black
and white RVs!)
0/1
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Expectation and Variance in Binomial Random Variables
 So easy!!!