Transcript Probabilities of common Games

Probabilities of common
Games
How do I avoid bad bets?
What is P (Win) ?
• 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.
number of ways to win
P(Win) 
number of possibilities
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?
Solutions
18
P (Black) 
37
18
P (Odd) 
37
1
P (0) 
37
18 18 18 18 104976
P (4red)     
 5.6%
37 37 37 37 1874161
18
P (Red) 
(It doesn't matter what happened
37
on previous rolls.)
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 number
means 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 1
way 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?
Solutions
6
2
8 2
P(win)  P(7)  P(11) 


  22%
36 36 36 9
1
2 1
4 1
P(lose)  P(2)  P(3)  P(12) 
 

  11%
36 36 36 36 9
4 1
P(win after 5)  P(5) 
  11%
36 9
6 1
P(lose after 5)  P(7) 
  18%
36 6
P(keep playing)  1  P(win or lose)  1  ( P(5)  P(7))
6 
10 26 13
 4
 1     1

  72%
36 36 18
 36 36 
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?
1. No!
Solutions
number of winning #'s 6
2. P(1#) 

possible numbers
49
3. 5 winners left and 48 total numbers left.
6 5
30
4. P(2 # s)  P(1st )  P(2nd )   
 1.27%
49 48 2352
6 5 4 3 2 1
5. P(6 # s)      
49 48 47 46 45 44
720
1


 0.000007%
10068347520 13983816
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 buying
a 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.
Solutions
10
1
1
P(10win)  P(win)    
60466176
6
10
10
9765625
5
P(10loss)  P(loss)    
 16%
60466176
6
10