Introduction to Discrete Probability - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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1

Introduction to Discrete

Probability

2

Terminology

• Experiment

• A repeatable procedure that yields one of a

given set of outcomes

• Rolling a die, for example

• Sample space

• The range of outcomes possible
• For a die, that would be values 1 to 6

• Event

• One of the sample outcomes that occurred
• If you rolled a 4 on the die, the event is the 4

4

Probability is always a value

between 0 and 1

• Something with a probability of 0 will never

occur

• Something with a probability of 1 will

always occur

• You cannot have a probability outside this

range!

• Note that when somebody says it has a

“100% probability)

• That means it has a probability of 1

5

Dice probability

• What is the probability of getting “snake-eyes”

(two 1’s) on two six-sided dice?

• Probability of getting a 1 on a 6-sided die is 1/
• Via product rule, probability of getting two 1’s is the probability of getting a 1 AND the probability of getting a second 1
• Thus, it’s 1/6 * 1/6 = 1/
• What is the probability of getting a 7 by rolling

two dice?

• There are six combinations that can yield 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
• Thus, |E| = 6, |S| = 36, P(E) = 6/36 = 1/

7

Poker probability: royal flush

• What is the chance of

getting a royal flush?

• That’s the cards 10, J, Q, K, and A of the same suit
• There are only 4 possible

royal flushes

• Possibilities for 5 cards: C(52,5) = 2,598,
• Probability = 4/2,598,960 = 0.
  • Or about 1 in 650,

8

Poker probability: four of a kind

• What is the chance of getting 4 of a kind when

dealt 5 cards?

• Possibilities for 5 cards: C(52,5) = 2,598,
• Possible hands that have four of a kind:
• There are 13 possible four of a kind hands
• The fifth card can be any of the remaining 48 cards
• Thus, total possibilities is 13*48 = 624
• Probability = 624/2,598,960 = 0.
• Or 1 in 4165

10

Poker probability: full house

• What is the chance of getting a full house? - That’s three cards of one face and two of another face
• We must do ALL of the following:
  • Pick the face for the three of a kind: C(13,1)
  • Pick the 3 of the 4 cards to be used: C(4,3)
  • Pick the face for the pair: C(12,1)
  • Pick the 2 of the 4 cards of the pair: C(4,2)
• As we must do all of these, we multiply the values out (via the product rule)
• This yields
• Possibilities for 5 cards: C(52,5) = 2,598,
• Probability = 3744/2,598,960 = 0.
  • Or about 1 in 694

11

Inclusion-exclusion principle

• The possible poker hands are (in increasing order):
  • Nothing
  • One pair cannot include two pair, three of a kind, four of a kind, or full house
  • Two pair cannot include three of a kind, four of a kind, or full house
  • Three of a kind cannot include four of a kind or full house
  • Straight cannot include straight flush or royal flush
  • Flush cannot include straight flush or royal flush
  • Full house
  • Four of a kind
  • Straight flush cannot include royal flush
  • Royal flush

13

Poker hand odds

• The possible poker hands are (in increasing

order):

• Nothing 1,302,540 0.
• One pair 1,098,240 0.
• Two pair 123,552 0.
• Three of a kind 54,912 0.
• Straight 10,200 0.
• Flush 5,108 0.
• Full house 3,744 0.
• Four of a kind 624 0.
• Straight flush 36 0.
• Royal flush 4 0.

14

More on probabilities

• Let E be an event in a sample space S. The

probability of the complement of E is:

• Recall the probability for getting a royal flush is

• The probability of not getting a royal flush is 1-0.0000015 or 0.
• Recall the probability for getting a four of a kind

is 0.

• The probability of not getting a four of a kind is 1-0.00024 or 0.

p ( E ) = 1 − p ( E )

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Probability of the union of two

events

S

E 1 E 2

p ( E 1 U E 2)

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Probability of the union of two

events

• If you choose a number between 1 and

100, what is the probability that it is divisible by 2 or 5 or both?

• Let n be the number chosen
  • p (2| n ) = 50/100 (all the even numbers)
  • p (5| n ) = 20/
  • p (2| n ) and p (5| n ) = p (10| n ) = 10/
  • p (2| n ) or p (5| n ) = p (2| n ) + p (5| n ) - p (10| n )

19

When is lotto worth it?

• Many older lotto games you have to choose 6

numbers from 1 to 48

• Total possible choices is C(48,6) = 12,271,
• Total possible winning numbers is C(6,6) = 1
• Probability of winning is 0.
  • Or 1 in 12.3 million
• If you invest $1 per ticket, it is only statistically

worth it if the payout is > $12.3 million

• As, on the “average” you will only make money that way
• Of course, “average” will require trillions of lotto plays…

20

Powerball lottery

• Modern powerball lottery is a bit different
  • Source: http://en.wikipedia.org/wiki/Powerball
• You pick 5 numbers from 1-
  • Total possibilities: C(55,5) = 3,478,
• You then pick one number from 1-42 (the powerball)
  • Total possibilities: C(42,1) = 42
• By the product rule, you need to do both
  • So the total possibilities is 3,478,761* 42 = 146,107,
• While there are many “sub” prizes, the probability for the jackpot is about 1 in 146 million - You will “break even” if the jackpot is $146M - Thus, one should only play if the jackpot is greater than $146M
• If you count in the other prizes, then you will “break even” if the jackpot is $121M