More Probability
LSP 121
LSP 121
Math and Tech
Literacy II
More Probability
Greg Brewster
DePaul University
Topics
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More Probability
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Combining Probabilities
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AND – multiply probabilities
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OR – add probabilities
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Average trials to success
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“At Least Once” calculations
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Expected Values
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What’s the payoff?
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Should you ever play the lottery?
Combining Events
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Sometimes you want to determine
the probability of a combination of
two event outcomes.
Given outcomes A and B
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What is the probability that EITHER A
OR B happen = Pr(A or B)?
What is the probability that BOTH A
AND B happen = Pr(A and B)?
Greg Brewster, DePaul University
Page 1
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LSP 121
Either/Or Probabilities Mutually Exclusive Events
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If events A and B are mutually exclusive then
P(A or B) = P(A) + P(B)
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Mutually exclusive means that these two
event outcomes cannot both happen.
Either/Or Probabilities
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Example: Suppose you roll a single die.
What is the probability of rolling either
a 2 or a 3?
P(2 or 3) = P(2) + P(3) = 1/6 + 1/6 = 2/6
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These events cannot both happen. A
single roll can’t be both a 2 and a 3, so
the equation works.
Either/Or Probabilities
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Example: Suppose you draw a card
from a deck of cards. What is the
probability that it is either a King or a
Queen?
P(K or Q) = P(K) + P(Q) = 4/52 + 4/52 =
8/52 = 15.4%
Greg Brewster, DePaul University
Page 2
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LSP 121
Either/Or Probabilities
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Example: Suppose you roll two dice.
What is the probability that either the
first roll is a 2 or the second roll is a 3?
We cannot just add these probabilities
together because it is possible that both of
these events occur. We need the following
more general equation:
P(A or B) = P(A) + P(B) – P(both A and B)
Either/Or Probabilities
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Example: Suppose you roll two dice.
What is the probability that either the
first roll is a 2 or the second roll is a 3?
P((1st roll is 2) or (2nd roll is 3))
=P(1st roll is 2) + P(2nd roll is 3)
– P(1st roll is 2 AND 2nd roll is 3)
= 1/6 + 1/6 – 1/36
= 11/36 = 30.55%
AND Probabilities for
Independent Events
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If events A and B are independent then
P(A and B) = P(A) x P(B)
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Independent means that the outcome of
event A has no bearing on the outcome of
the event B.
Greg Brewster, DePaul University
Page 3
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LSP 121
Calculating Probabilities Independent Events
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For example, suppose you toss three
coins. What is the theoretical probability
of getting three tails?
1/2 x 1/2 x 1/2 = 1/8 = 12.5%
Calculating Probabilities Independent Events
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A 100-year flood is a flood that occurs, on
average, only once every 100 years.
Find the theoretical probability that a 100year flood will strike a city in two
consecutive years
(1 in 100) x (1 in 100) = 0.01 x 0.01 =
0.0001 = 0.01%
Calculating Probabilities Independent Events
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The Pick-Three game in the Illinois lottery is a game is a
game where you must correctly match a 3-digit number
to win.
What is the probability that you win the Pick-Three game
on a random try?
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P(win) = P((1st digit correct) AND (2nd digit
correct) AND (3rd digit correct))
P(win) = P(1st digit correct) x P(2nd digit
correct) x P(3rd digit correct)
= 1/10 x 1/10 x 1/10
= 1/1000 = 0.001 = 0.1%
Greg Brewster, DePaul University
Page 4
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LSP 121
Average Tries Before Success
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If you are repeatedly doing an activity
(trials) and each repetition is independent
and the probability of succeeding on each
trial is p, then
The average number of trials before you
succeed is 1/p
Average Tries Before Success
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On average, how many times do you need
to play the Pick-Three game before you
will win?
Answer: the probability of success each
time you play is p = 0.001. So the
average number of times you need to play
before you win is 1/p = 1000 times
Average Tries Before Success
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On average, how many times do you need
to roll a die before you get a 3?
Answer: the probability of rolling a 3 on
each try is p = 1/6. So the average
number of times you need to roll before
you roll a 3 is 1/p = 6 times
Greg Brewster, DePaul University
Page 5
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LSP 121
Probability of At Least One
Success in N Trials
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What is the theoretical probability of
something happening at least once if
you try it N times?
P(at least one A in N trials) =
1 - [P(not A in one trial)]N
Probability of At Least Once Example
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What is the probability that you will win the PickThree game at least once if you play 100 times?
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Probability of winning each time is 1/1000.
Probability of not winning each time is (1 – 1/1000) =
999/1000.
P(at least one win in 100 tries) =
1 - 0.999100 = 0.095 or 9.5%
How about if you play 1000 times?
 P(at least one win in 1000 tries) =
1 - 0.9991000 = 0.632 or 63.2%
Probability of At Least Once Example
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What is the theoretical probability that a region
will experience at least one 100-year flood
during the next 100 years?
Probability of a flood is 1/100.
Probability of no flood is 99/100.
P(at least one flood in 100 years) =
1 - 0.99100 = 0.634
Greg Brewster, DePaul University
Page 6
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LSP 121
Probability of At Least Once
Another Example
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You purchase 10 raffle tickets, for which the
theoretical probability of winning some prize
on a single ticket is 1 in 10.
What is the theoretical probability that you
will have at least one winning ticket?
P(at least one winner in 10 tickets) =
1 - 0.910 = 0.651
Expected Value
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Now, what if we assign values to the
outcomes we are working with?
For example:
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Playing the Pick-Three game costs $1
Winning the Pick-Three game gives you $500
Then we can compute the expected value
(also called the “expected winnings” or
“expected losses”) of a game.
Calculating Expected Value
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You compute the expected value by
going through all the possible
outcomes and adding up the (value x
probability) for each outcome.
Expected value =
outcome 1 value x outcome 1 probability
+ outcome 2 value x outcome 2 probability
+…
Greg Brewster, DePaul University
Page 7
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LSP 121
Expected Value of Pick-Three
Game
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The outcome values for the Pick-Three
game are the following:
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Winning the game has value $499 (costs $1 to play
and $500 winnings) and probability 0.001
Losing the game has value -$1 (costs $1 to play and
no winnings) and probability 0.999
Expected value = $499 x 0.001 + (-$1 x 0.999)
= -$0.50 = 50 cent loss per play
Over the long run, on average, you expect to lose 50
cents per play. Play 100 times => lose $50 average
Another Example
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Suppose that $1 lottery tickets have
the following probabilities:
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1
1
1
1
in
in
in
in
5 win a free $1 ticket;
100 win $5;
100,000 to win $1000;
10 million to win $1 million.
What is the expected value of a lottery
ticket?
Example I - Solution
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Ticket purchase: value -$1, prob 1.0
Win free ticket: value $1, prob 1/5
Win $5: value $5, prob 1/100
Win $1000: prob 1/100,000
Win $1million: prob 1/10,000,000
-$1 x 1= -1;
$1 x 1/5 = $0.20;
$5 x 1/100 = $0.05;
$1000 x 1/100,000 = $0.01;
$1,000,000 x 1/10,000,000 = $0.10
Greg Brewster, DePaul University
Page 8
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LSP 121
Example I - Solution Continued
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Now sum all the products:
-$1 + 0.20 + 0.05 + 0.01 + 0.10
= -$0.64
Thus, averaged over many tickets, you should expect
to lose $0.64 for each lottery ticket that you buy.
If you buy 1000 tickets, you should expect to lose
$640, on average.
Example II
The Insurance Business
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Suppose an insurance company sells policies for
$500 each.
The company knows that 10% of their clients will
submit a successful claim that year and that
successful claims average $1500 each.
How much can the company expect to make per
customer?
Insurance Example
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Company makes $500 100% of the time
(when a policy is sold)
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Company loses $1500 10% of the time
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($500 x 1.0) + ( - $1500 x 0.1) = 500 – 150 = $350
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Company gains $350 from each customer it sells a
policy to, on average.
Greg Brewster, DePaul University
Page 9
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LSP 121
Does it ever make
sense to play Lotto?
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The Illinois Lottery “Lotto” game requires you to
match six numbers.
Odds of winning are 1 in 10,179,260 (according to
their web site). So, probability of winning each time
is p=1/10,179,261.
The Jackpot you win changes all the time
Expected value per play = -$1 + p * Jackpot
So, if the Jackpot is larger than $10,179,261, then
the expected value is greater than zero – all right!
But, on average, you need to play 1/p = 10,179,261
times before you win. I wouldn’t recommend it.
Greg Brewster, DePaul University
Page 10