Transcript Probability in gambling

Probability in gambling
Lucie Tejessyová
Gambling
• Games of chances or skills when betting
ratios are to bettors disadvantage
• The best bets – bring the casino the smallest
gain
• Their outcome depends mainly on the
chance
Probability
• Probability theory – mathematical discipline
that analyses random events
• Random event – a result of a random
experiment
• Random experiment – its course and
outcome depend on the chance
Classical definition
of probability
• The number of all results is final
• All results are equally possible
• All results are mutually exclusive
• The probability of an event 𝐴 is defined as
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑠𝑒𝑠 𝑓𝑎𝑣𝑜𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑒𝑣𝑒𝑛𝑡 𝐴 𝑚
𝑃 𝐴 =
=
𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑐𝑎𝑠𝑒𝑠
𝑛
Odds & expected value
• Odds – the probability that event A occurs to
the probability that the event A does not occur
𝑃(𝐴)
O𝑑𝑑𝑠 =
1 − 𝑃(𝐴)
• Expected value – the mean estimate of future
outcome
𝐸(𝑥) = 𝑥1 ∙ 𝑝1 + ⋯ + 𝑥𝑛 ∙ 𝑝𝑛
Roulette
• Two versions – french (0) and american (00)
• The payout ratio can be calculated by
1
36 − 𝑛 where 𝑛 is the number of squares
𝑛
the player is betting on
• If the 0 comes up, all bets which have not been
placed on zero, lose instantly
• Even-money bets – red x black, high x low,
even x odd
French Roulette
Bet
Probability
Odds
Payout
True Math.
House
Odds
Advantage
Straight
0,027
1 ∶ 36
35 ∶ 1
36 ∶ 1
2,7 %
Split
0,054
2 ∶ 35
17 ∶ 1
35: 2
2,7 %
Street
0,081
3 ∶ 34
11 ∶ 1
34: 3
2,7 %
Corner
0,108
4 ∶ 33
8∶1
33: 4
2,7 %
Six line
0,162
6 ∶ 31
5∶1
31: 6
2,7 %
Dozens
0,324
12 ∶ 25
2∶1
25: 12
2,7 %
Even-money
0,486
18 ∶ 19
1∶1
19: 18
2,7 %
Sportka
• Lottery game
• The aim is to guess 6 numbers out of 49
•𝑃=
6
𝑛
43
∙ 6−𝑛
49
6
, where 𝑛 is the number of
correctly guessed numbers
(At second prize multiply by
1
)
10
• Profit of Sazka - 50 % of all deposits
Probability of win in Sportka
Prizes
Number of guessed
numbers
Probability
Winning
percentages
1. Prize
6
0,00000715 %
22 %
2. Prize
5 + 1 additional
0,0000429 %
7%
3. Prize
5
0,001845 %
9%
4. Prize
4
0,09686 %
12 %
5. Prize
3
1,765 %
40 %
Loss
0–2
98,14 %
0%
Conclusion
• The casinos have always certain probability
advantage above players and therefore they are
definitely profitable
• None of these games is fair
• The probability of most wins is less than 5% and
therefore it is practically impossible to reach
them
• Gambling should be used for entertainment, not
to earn money
„The greatest advantage in gambling
lies in not playing at all.“
(Girolamo Cardano)