Batch 3 - Class 257 - Probability in Gambling Blackjack

• In a diwali mela, I have a game stall. I draw two cards from a deck of cards. You draw a third card and if that third card lies between my two cards, you win. Assume that all cards are ordered, so that no two cards have the same value (i.e. we have a tiebreaker mechanism, like the suit color). What odds would you be willing to take to play a fair game?

• 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

• Suppose we both draw a card from a deck, and the higher card wins. What is the probability that you will win? 50%
• Suppose I showed you my card before you bet. What will you do and what will be your probability of winning?
• Now you can bet or not depending on my card (non-independent events) - let students construct some simple betting strategy - they can choose to bet or not, or bet different amounts. See if this improves their expected value
• Conditional probability - example, chances of drawing an ace and king in succession in a deck of  cards?

• Each player is dealt two cards, and the dealer is dealt two cards, one of which is face up. 10/J/Q/K have value of 10, 2-9 have card value, and Ace can count as 1 or 11. The goal is to get as close to 21 as possible without exceeding it (bust). First all players play. Then the dealer plays, and can't hit at/above 17 (or can "soft hit" at 17, i.e. hit if there is an Ace there being counted as 11). 
• Play a few games
• What are the chances of "two card blackjack"?
• 4/52*16/51*2=4.82% (Note that this is 1/21 - the total we are trying to reach is also 21. The game is also called 21!)
• Suppose you get a 8 and a 6. Will you ask for a third card? Assume the dealer has 4.
• The probability of dealer going bust with a 4 card? (Happens to be 39.6% but how do we think about it). How do you program that?
• Simulations as an alternate strategy to computation
• Now we will compute probability of winning by drawing a card versus not. Which one do you think is better?
• If you draw A,2,3,4,5,6,7 you are better off. That is 26 possible cards
• Worse off in 8,9,10,J,Q,K i.e. 23 possible cards
• So you draw
• Who do you think has advantage in an overall blackjack game? Why
• Note that the dealer has strict rule to stand on 17, but that is in any case a "good decision". Since the player busts before the dealer needs to, there is a slight house advantage. With a single deck of cards, there is a 0.17% house advantage.
• Again, we can run simulations of thousands of games
• Probability of win 41.4%, tie 9.5%, loss 49.1%
• If both player and dealer busting was a tie, then we would have 41.4%, 17.1% and 41.5% - that one rule makes all the difference!
• If you play with two decks instead of one, should the house advantage increase or decrease?
• 1/13*32/103*2=4.78% - probability of player blackjack reduces. Similarly, probability of tie decrease. 
• House advantage goes from 0.17% for single deck, to 0.46% for two decks and 0.66% for eight decks
• How should we think about strategy to play blackjack?
• Our play should be dependent on the card that dealer has shown and our own set of cards
• At this moment we assume no card counting, i.e. no knowledge of previous hands
• What are we trying to optimize? Expected value. So if we get a 2x on a win, then at every stage, we are trying to look at probability of increasing our chances of win.
• Note that this is a "sunk cost" game, so you have already bet the money when you started playing
• If we have an "soft Ace" the play can be different due to optionality - generally, you are going to hit more if you have a soft Ace. 
• If the totals are high, you are likely to stand. If the total is low you are likely to hit.
• If the dealer has very small or very high cards, then you play conservatively. Middling cards are not as good. 

• Card Counting: Some of the most famous exploits on blackjack have been with card counting. So as the deck is being dealt for successive plays, one can count the cards, to predict the outcomes of future draws - remember, "conditional probability"
• Card counting can alter the house advantage from 0.5-1% to negative 2%!
• Suppose in a single deck system, you have seen 4 10/J/Q/K s being player in last 4 hands where total of 20 cards played. You have a 5 and 6. The dealer is showing a 10. How do your chances of a "3 card 21 total) change depending on whether you are counting cards or not?
• 11/28 versus 15/49 - this can translate to distinctive advantage
• Normally, you dont count every card, but maintain a running "count" depending on a range of cards (1 for values 2-6, 0 for 7-9, -1 for 10-A) and use that to change the play table.
• See movie "21" about MIT's blackjack team that 
• Casinos do various things to prevent card counting - ban players, increase decks, shuffle more often and so on
• How should you play in a casino
• Suppose you are playing a "fair game" with 50:50 chances and a 2x payout. You walk into the casino with $100. What are you likely to walk out with?
• Depends on the number of games - you can win a lot, or you can be bankrupt
• How do you think the chances of going bankrupt change as you play more games?
• How do you think the maximum amount you can win changes as you play more games?
• As an exercise, lets assume bets of $25. And lets say we play 3,4,5,6 games. Let students compute chances of going bankrupt and maximum amount they can win (separate students can do separate sections)
• Note that both maximum amount, but also chances of going bankrupt increase as you play more. Why does the latter happen?
• The casino never goes bankrupt and hence has a natural advantage, regardless of odds on a game (well, they also have an advantage on the game itself!)
• If you want a "fair game" you have to limit your winnings. So if you can afford to lose upto $100 fairly, how much should you be willing to win "fairly". 
• $100. That means that if you win $100, you take the money and leave. The more you are trying to win, the more you are increasing your chances of a bust
• Of course, once you have already won $100, then conditional probability applies, and you can win more :-)

• 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.