Batch 3 - Class 258 - Probability in Gambling (Bookmakers bets)
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Pre-Class Exercise
A CALENDAR PUZZLE. If the end of the world should come on the first day of a new century, can you say what are the chances that it will happen on a Sunday?
Answer: The first day of a century can never fall on a Sunday; nor on a Wednesday or a Friday.
Independent events: History does not influence the future outcomes. Eg: roll of dice
Probability: For independent events, probability of successive outcomes is product of individual probabilities
Equally likely events
Expected outcomes: Win/loss multiplied by respective probabilities
House Advantage in gambling - 5.26% for roulette
Volatility Index - players can still win even though for an infinite play, the house has an advantage
Strategies to beat the house - uncover the bias, predict (identify dependent events like early position of the roulette ball), Martingale strategy
We have discussed Roulette and Blackjack so far. Today, we will create our own game.
What should be the criteria for a good game design? (Instructor Notes: Let kids come up with this)
Easy to explain - say, should be possible to explain in under 30 seconds
Easy to play - not too many side bets, variations etc
House advantage - how much?
Should seem fair
Build upon an existing game - familiarity
Speed of playing - allows players to play many times in a shorter period of time
Visually attractive...
We know of some base starting points - roulette, cards, dice
So lets say both the player and the dealer toss a coin. Whoever gets heads wins. In case of a tie, house wins
What criteria does it satisfy?
What is the house advantage? Is it too high/too low (Instructor notes: let students compute)
Can you figure out ways of making the game more attractive to players
For example, what if in both heads, the amount was split - how does the house advantage change
Or, in case of tie, both players roll again and if the player wins, she gets back twice of the total amount bet (in both rolls)
Other ideas by which you can make it more interesting?
Let students now design their own game, and explain to other students. Let other students vote on whether they would play or not, and balance it against house advantage (2x2 plot)
Counter-intuitive probability - let students compute
Example: The game we saw last class - drawing three cards and the player wins if the third card is in middle of first two - simply, but not immediately obvious (1/3 chance of player win)
Example: I will let you nominate three cards, and I will then draw three cards - you win if none of the cards you nominate are drawn (45% chance of player win) - my cards drawn may be with out without replacement
Example: In a group of 30 people, probability of (at least) two people having the same birthday is 70%
Example: Pick a number between 0 to 9, and if I guess your number correctly, I win 10x (people tend to pick 3 and 7 disproportionately)
Bookmakers
Bookmakers are people who run a betting book. They are themselves not interested in taking a position, but just want to make a margin for providing this service. How do they manage their risk?
For example, lets say Nadal is playing Federer, and the expected chances for Nadal winning are higher. Then the bookies may offer lower reward for a bet on Nadal (say 1.5 for each rupee bet), and a higher for Federer (say 2.5 for each rupee bet).
Note that the betting payouts are higher for events with lower probability
In this arrangement, the bookie doesn't want to take any risk - the "margin" that the bookie has here is 1/1.5+1/2.5 = 1+8%. 8% is called the margin
However, even with this, if there were 100 bets each on Nadal and Federer, and Federer won, then the bookie would have collected Rs 200 against the 200 bets, but has to payout Rs 250 to everyone who won by betting on Federer.
What would the bookie do if they wanted to make sure that they always make their 5% margin?
Since they want to distribute Rs 190, ideally they should have 190/2.5 ~ 75 bets for Federer, and 190/1.5 ~ 125 bets for Nadal
Then regardless of who wins, the bookie makes Rs 10
Given that the bookie doesn't know in advance how people will bet, how should they adjust to the betting pattern so that they don't loose money?
Lets say that they start with an assumption that equal number of people bet on Nadal and Federer, and offer the same 1.9 payout for each win
Now the bookie observes that 1000 people have bet on Nadal and only 940 on Federer. At this stage
1940 is collected as bets
If Nadal wins, the payout is 1900, and the bookie makes only Rs 40 (lower than Rs 100 target)
If Federer wins, the payout is ~1800, and the bookie makes Rs 140, which is higher than target
However, if this keeps happening, with 2000 bets on Nadal and 1880 on Federer, then the margin keeps narrowing and the bookie doesn't want that
So the bookie can now raise the payout for Federer to 2 and lower the payout for Nadal to 1.8. If now he gets incremental 1000 bets on Nadal and 940 on Federer
Total collections 3880
Nadal wins 1000*1.9+1000*1.8=3700 - bookie makes 1880
Federer wins 940*1.9+940*2 ~ 3700 - bookie makes 1880
Hence by "rebalancing the bets" the bookie has been able to adjust the odds and ensure his earnings - by lowering the payout where more bets are being put, and increasing the payout where fewer bets are being put.
So the bookmakers are not predicting the win, but if more people are betting on a particular option, the odds offered for that option go down.
Note that balancing of books may also attract more bettors towards the currently unfavoured option, and hence help balance the book further
What keeps the bookmakers honest and not offer low odds? Competition - the customer always has an option to take bets from a different bookmaker who is more fair.
Another solution is for different bookmakers to establish a wholesale exchange - since different bookmakers might be seeing different kinds of bets, they can balance it out through the wholesale exchange.
Homework:
Place 52 cards from a deck face down in a line. Turn the first card, and then move number of cards further equivalent to the face value of the opened card. Now open the card that you land on and continue the same process till you get to the end of the deck (meaning the last card, beyond which when you try to move by face value of the card, you would run out of cards). Mark the last card that was opened. Now, if I do the same process, starting not from the first card but any of the next 9 cards (card numbers 2-10), what are the chances of my last card being the same as yours?
Answer: about 80%. Note that on each step there is a 1/13 chance (actually more, since you may have two of your opened cards in my next 13 cards on an average) that I land up on one of your prior opened cards, and then I follow the same journey. In a 52 card stretch, this opportunity may arise about 7 times. That combines to give fairly high chances that one would land up on the same card at least once.
Roughly 2/13+11/13*(2/13+11/13*(2/13+11/13*(2/13+11/13*(2/13+11/13*(2/13+11/13*(2/13)))))) ~ 69% (actual probability is around 80%)