Probabilities of common Games

How do I avoid bad bets?

What is• Recall: The probability of an event is the

number of ways it can happen (successes) divided by the total number of possible outcomes.

• For a game we are interested in winning, the probability of a win is the number of ways to win divided by the total number of possibilities.

(Win)?P

number of ways to win(Win)

number of possibilitiesP

Roulette

• Roulette is played by rolling a ball around a spinning wheel. The ball will eventually fall in one of the numbered pockets.

• The pockets are numbered 0 to 36.

• From 1 to 36 half are red and half are black.

• 0 is green, and is always a loser.

• Players bet on specific numbers,colours, or groups of numbers.

Figure it out!

• What is the probability of getting black?

• What is the probability of getting odd?

• What is the probability of getting 0?

• What is the probability the wheel will come up red 4 times in a row?

• What is the probability the wheel will come up red if it just came up red 4 times in a row?

Solutions18

(Black)37

18(Odd)

371

(0)37

18 18 18 18 104976(4red) 5.6%

37 37 37 37 187416118

(Red) (It doesn't matter what happened 37

on previous rolls.)

P

P

P

P

P

How does it work?

• The “House” (casino) advantage comes from the green 0 slot.

• E.g. Betting on black would double your money if black comes up, but because of the green 0, black will come up slightly less than half the time.

• American roulette has even worse probabilities because there is another green slot with a 00.

Craps

• Craps is a game played by rolling 2 dice.

• The first roll of a round has 3 outcomes: win lose or pass.

• A roll of 7 or 11 wins.

• A roll of 2,3 or 12 loses.

• Any other numbermeans the player has to roll that number again before rolling a 7 to win.

What is it about 2 dice?

• Add up the 2 dice to get the total.

• Different results have different probabilities.

• E.g. There is only 1way to get a 2 (1,1)but 3 ways to get 4(1,3);(2,2);(3,1).

Solve it!• What is the probability of winning right

away?• What is the probability of losing right away?• Suppose you roll a 5.

-What is the probability you win on the next roll? -What is the probability you lose on the next roll?-What is the probability you have to roll again?

Solutions6 2 8 2

(win) (7) (11) 22%36 36 36 9

1 2 1 4 1(lose) (2) (3) (12) 11%

36 36 36 36 94 1

(win after 5) (5) 11%36 96 1

(lose after 5) (7) 18%36 6

(keep playing) 1 (win or lose) 1 ( (5) (7))

4 61

36 36

P P P

P P P P

P P

P P

P P P P

10 26 13

1 72%36 36 18

How does it work?

• Gambling on craps is done by betting on different outcomes (wins or losses)

• The house (casino) sets the payouts so that it is unfavourable for the player.

• E.g. Because a 2 (1,1) has a 1 in 36 chance of happening, winning a bet should pay 36 times the $ amount but actually only pays 31 times.

Lotto 649

• Lotto 649 is a large Canadian lottery.

• Players select 6 different numbers from 1 up to 49. The order does not matter.

• The jackpot is awarded to players whose 6 numbers match the 6 winning numbers

• Lesser prizes are awarded for matching some of the numbers.

Solve it!• Does this game allow repetition?• What is the probability of selecting one of

6 winning numbers from 1 to 49?• After selecting a winning number, how

many winners are left? How many total numbers are left?

• What is the probability of selecting 2 winning numbers from 1 to 49?

• What is the probability of selecting 6 winning numbers from 1 to 49?

Solutions1. No!

number of winning #'s 62. (1#)

possible numbers 49

3. 5 winners left and 48 total numbers left.

6 5 304. (2# ) (1 ) (2 ) 1.27%

49 48 23526 5 4 3 2 1

5. (6 # )49 48 47 46 45 44

720 1

10068347520 139838

P

P s P st P nd

P s

0.000007%16

How does it work?• The jackpot payout in 649 is very large but the

probability of winning is very small.• At $2 per ticket, a player would have to spend

almost $28 million to be guaranteed of winning the jackpot.

• The jackpot is divided among the winners.• Because the odds are so small, people don’t pay

attention; our minds are not good at the difference between “1 in a million”, “1 in 14 million” and “a long shot”.

• $2 per ticket is considered a small risk.

Roll up the Rim to Win!

• Every Spring Tim Horton’s coffee cups have a chance of winning a prize.

• The prizes have different probabilities ranging from 1 in 6 for a coffee to 1 in 11353500 for a Rav4 car.

• The game is free to play(if you are already buyinga coffee).

Ponder it!

• Why can we assume the probability for winning a prize is 1 in 6?

• Is Roll up the Rim to Win with or without replacement? How would this affect the probabilities?

• What is the probability of winning 10 times in a row?

• What is the probability of losing 10 times in a row?

Solutions

• Because there are so many more coffees to be won, considering the other prizes hardly changes the probability at all!

• RUTRTW is without replacement, but because there are so many cups, taking one and not replacing it does not significantly affect the probability. It would be important to consider replacement with the expensive prizes because there are so few of them.

Solutions10

10

1010

1 1(10win) (win)

6 60466176

5 9765625(10loss) (loss) 16%

6 60466176

P P

P P