Probability for powerball and poker
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Probability for Powerball and Poker. Extra Chapter 4 stuff by D.R.S., University of Cordele. 2 of these and 4 of those. A classic type of problem You have various subgroups. When you pick 6, what is the probability that you get 2 of this group and 4 of that group?

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Probability for Powerball and Poker

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Probability for powerball and poker

Probability for Powerball and Poker

Extra Chapter 4 stuff

by D.R.S., University of Cordele


2 of these and 4 of those

2 of these and 4 of those

  • A classic type of problem

  • You have various subgroups.

  • When you pick 6, what is the probability that you get 2 of this group and 4 of that group?

  • Jellybeans: 30 red, 30 yellow, 40 other

  • Choose 6. Find P(2 red and 4 yellow)


2 red out of 30 4 yellow out of 40

2 red out of 30; 4 yellow out of 40

  • Analysis – you must THINK! – “This is a Fundamental Counting Principle situation…

    • One event is drawing 2 red out of 30

    • The other is drawing 4 yellow out of 40

    • FUNDAMENTAL COUNTING PRINCIPLE says to multiply how many ways for each of them.

    • Each of these events is modeled by a COMBINATION, because the order doesn’t matter.

  • So how do you write it in Combination language?


Computing the probability

Computing the Probability

  • Jellybeans: 30 red, 30 yellow, 40 other

  • Choose 6. Find P(2 red and 4 yellow)

  • Always go back to

  • Numerator:

  • Denominator:


Exactly aces

“Exactly aces”

  • Draw 5 cards, what is the probability of exactly 0 aces?

  • We can do this with our earlier techniques:

    • P(first card not at ace) = ____ / 52, times …

    • P(second card not an ace) = ____ / 51, times …

    • P(third card not an ace) = ____ / 50, times …

    • P(fourth card not an ace) = ____ / 49, times …

    • P(fifth card not an ace) = ____ / 48


Exactly aces1

“Exactly aces”

  • P(0 aces out of 5 cards drawn)

  • A more sophisticated view

    • 5 non-aces out of 52 cards

    • How many non-aces are there?

  • Numerator: ways to get 5 non-aces:

  • Denominator: total 5-card hands:

  • P(0 aces) =


Exactly aces2

“Exactly aces”

  • P(exactly 1 ace out of 5 cards drawn)

  • Our earlier techniques could do P(≥1 ace)

  • But P(=1 ace) would be harder or impossible

  • Counting techniques makes it easier

    • Choose 1 ace out of 4 aces

    • Choose 4 other cards out of 48 non-aces

  • P(1 ace) =


Exactly aces3

“Exactly aces”

  • Similiarly for 2 aces, 3 aces, 4 aces:

  • P(2 aces) =

  • P(3 aces) =

  • P(4 aces) =

  • Check: P(0) + P(1) + P(2) + P(3) + P(4) must total to exactly 1.000000000000000000. Why?


Probability of a full house

Probability of a Full House

  • Three of a kind

    • Choose 1 out of 13 ranks

    • Choose 3 out of 4 suits

  • One pair

    • Choose 1 out of the remaining 12 ranks

    • Choose 2 out of the 4 suits

  • P(full house) =


Probability of a flush

Probability of a Flush

  • A Flush: five cards all of the same suit

    • Choose 1 out of the 4 suits

    • Take 5 out of the 13 ranks

  • P(flush) =


How many different powerball tickets can be composed

How many different Powerball tickets can be composed?

  • Choose 5 out of the 59 white numbers.

  • Choose 1 out of the 35 red powerball numbers.

  • The Fundamental Counting Principle: Multiply the number of outcomes of the sub-events.

  • There are therefore possible ways to play the ticket, not counting the extra PowerPlay “multiplier” option.


59 c 5 35 c 1

(59 C 5) ∙ (35 C 1)

  • Repeating: possible ways to play the ticket, not counting the extra PowerPlay “multiplier” option.

  • This is the number of outcomes in the sample space.

  • Therefore this is the denominator in each of our powerball probability calculations.


From powerball com web site

From powerball.com web site


Powerball jackpot

Powerball Jackpot

  • You choose 5 out of the 59 white numbers

    • All 5 match the 5 winners

  • You choose 1 out of the 35 red numbers

    • And it matches the winner

  • Numerator is

  • Denominator as before, (59 C 5)(35 C 1).

  • Compare this result to the odds printed on the ticket.


Powerball 200 000

Powerball $200,000

  • You choose 5 out of the 59 white numbers

    • All 5 match the 5 winners

  • You choose 1 out of the 35 red numbers

    • And it is one out of the 34 that don’t match the winner

  • Numerator is

    • Notice we still have 5 out of 5 on the white numbers

    • But the Powerball choice is 1 out of 34 losers

  • Reconcile this result with the printed odds.


Powerball 10 000

Powerball $10,000

  • You choose 5 out of the 59 white numbers

    • 5 winners but you picked got 4 of them

    • 54 losers and you picked one of those

  • You choose 1 out of the 35 red numbers

    • And it matches the winner

  • Numerator is

    • For the $100 via 4 white only with no red match, just change the to a


Powerball 7

Powerball $7

  • Two ways to win $7

  • 3 white matches, 2 losers; red is no match

  • Another way: 2 white matches, 3 losers, and the red powerball matches


Two ways to lose

Two ways to lose

  • Match absolutely nothing at all

    • 5 out of the 54 losing white numbers

    • 1 out of the 34 losing red powerball numbers

  • Or match 1 white number only

    • 1 out of the 5 winning white numbers

    • 4 out of the 54 losing white numbers

    • 1 out of the 34 losing red powerball numbers


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