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Probability for Powerball and Poker

Dive into the world of probability with this detailed examination of common scenarios in games like Powerball and Poker. From understanding combinations and counting principles to calculating the chances of drawing specific cards or winning a jackpot, this chapter unravels complex problems in an accessible way. Learn how to assess probabilities for various outcomes, such as drawing jellybeans or hands of cards, while gaining insight into critical thinking necessary for mastering these concepts. Essential for students and enthusiasts alike.

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Probability for Powerball and Poker

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  1. Probability for Powerball and Poker Extra Chapter 4 stuff by D.R.S., University of Cordele

  2. 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)

  3. 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?

  4. 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:

  5. “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

  6. “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) =

  7. “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) =

  8. “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?

  9. 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) =

  10. 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) =

  11. 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.

  12. (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.

  13. From powerball.com web site

  14. 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.

  15. 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.

  16. 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

  17. 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

  18. 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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