Probability for Powerball and Poker

Extra Chapter 4 stuffby 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?• 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

• 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

• Jellybeans: 30 red, 30 yellow, 40 other• Choose 6. Find P(2 red and 4 yellow)

• Always go back to • Numerator: • Denominator:

“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 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 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 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

• 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

• 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?

• 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)∙• 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

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

• 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

• 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

• 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

• 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