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# Probability for Powerball and Poker - PowerPoint PPT Presentation

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

Extra Chapter 4 stuff

by D.R.S., University of Cordele

• 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)

• 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?

• 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?

• 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) =

• 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) =

• 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.

• 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.

• 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.

• 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.

• 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

• 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

• 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