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Blue Cab/Green Cab
And Other Probability Calculations
 In chapter 2 of Flirting With Disaster, a variety of conditional probability problems are discussed, including one that describes the reliability of a courtroom witness identifying the driver in a hit-and-run collision under nighttime conditions. The original version of this problem was stated as one of recognizing the color of a taxi cab, perhaps to avoid the issue of race.
This page contains another approach to this problem using tables, a reliable -- albeit somewhat more analytical -- way to understand this type of problem. Here's the problem:
A witness to a hit-and-run accident claims that the vehicle that left the scene was a green taxi cab. In the town in question, 85 percent of the taxis are blue and 15 percent are green. The police establish that under the circumstances of the accident, the witness was likely to correctly identify the color of the cab 80 percent of the time. What is the likelihood that the witness' identification is correct?
 Most conditional probability problems of this type can be understood by creating a two-by-two table that contains all the relevant numbers. We know that the proportions of blue and green cabs are 85 percent to 15 percent, so that we can construct a hypothetical cab population of 1000 cabs to help fill in the rest of the numbers of interest.
Knowing that the witness gets the identification correct 80 percent of the time allows us to fill in the blue/blue and green/green cells, the upper left to lower right diagonal. The values are 680 and 120, respectively, .80 times each row total. By subtraction from the row totals, we can calculate that 170 blue cabs would be misidentified as green and 30 green cabs misidentified as blue. (You can also multiply the row totals by 1 minus 0.20.)
When you add up the columns, you can easily see how the counter-intuitive conclusion emerges.
If the witness were to identify the cab as blue, she would be correct 96 percent of the time (680/710), but if she were to identify the cab as green, she would be correct only 41 percent of the time (120/290) because the large number of "false positives" arising from the 20 percent error rate on the much larger blue cab population (.20 x 850 = 170) lowers the accuracy of her identification.
The equally disconcerting cancer test problem on page 39 of FWD can be unravelled with exactly the same type of analysis.
The Cancer Test
 A 98 percent accurate test produces the numbers at the left for a population of 10,000. The so-called "base rates" appear in the right most columns under Total. In this population of 10,000, 50 will have cancer, and 49 of them will test positive using the test (98 percent) and 9751 of the 9950 people who do not have cancer will test negative (also 98 percent).
By now you know that by looking at the other diagonal (lower left/upper right), you can identify the false negatives (1) and the false positives (199). The latter figure provides some hope for anyone who might receive some potentially bad news from their doctor, since they have only about a 20 percent chance of having cancer despite the results of the test. (Note that the chances of living in blissful ignorance despite having cancer is extremely small, only 1 in 9,752.)
Bringing Down the House
 Ben Mezrich's 2002 tale of the successful, perfectly legal team-based attack on the casinos of Las Vegas and other locales by six MIT students is as clever is it is audacious. Bringing Down the House is a wonderful read, as is the shorter version that appeared in Wired, and available on-line. (There is also a movie, "21," produced by Kevin Spacey, and several TV versions of the story.) Card counting stories are reminiscent of Robin Hood vs. the evil Sheriff.
Here, we deal with a bit of the math associated with card-counting, the conditional probability truth that allows a careful person with skill, patience, a large bankroll, and a willingness to become persona non grata in every casino whose security staff gets to know his or her face.
Unlike virtually all the other casino games, blackjack has a memory. As cards run through the shoe, the probability that particular cards will come up changes. Unlike roulette, whose odds remain constant, winning at blackjack depends on the cards that have been played since the last shuffle. When there are a greater number of face cards remaining to be played than at the start of the game, the chances that the dealer will "bust" increases, as does the advantage of other types of card play, such as doubling down, insuring, and splitting, that are not available to the house.
Of course, the casinos recognize their vulnerability. While card counting is technically legal in virtually all casinos, various means have been developed to prevent its successful practitioners. Dealers are taught to recognize the card counter's habits, especially the change in betting behavior when the count turns in their favor. Depending on the rules, casinos can initiate a shuffle, or in various other ways restore the odds to normal. In some cases, they even monitor the cards in the shoe before they are dealt in order to know which tables might be ripe for card counter play. In Las Vegas, casinos are private property, so they are allowed to bar any player, and will do so to recognized professionals that they see as a threat. In Atlantic City, the rules are different, but casinos have developed other strategies to restore their edge.
 Three Simulated Runs Through a Six-deck Shoe: The "True Count" Using the Hi-Lo Card Counting System
The graphs above show three selected simulations of the running "count" of cards as they go through a six deck shoe using the hi-lo card counting system that simply adds to the count for low numbered cards, subtracts from it for the high numbered cards (10s through aces), and ignores the middle valued cards 7, 8, and 9. The left graph shows a "hot shoe" that would encourage a card counter to make large bets when the count is high. The right hand graph is nearly its mirror image. In the middle is a run that stays neither high nor low until the very end. (The spiking at the end arises because the true count in the simulation is sensitive to the number of cards remaining to be played.)
When blackjack is played with flawless "basic strategy" (optimal play that ignores the count of the cards), the house edge is less than 0.5 percent. A card counter who ranges his bets appropriately will have an advantage of approximately one percent over casinos playing with six deck shoes. Such a slim advantage means that producing an acceptable profit can take hundreds of hours of play. The deck will only have a positive enough count for the player to raise his bets 10% to 30% of the time, depending on house rules, penetration into the deck, and the player's strategy.
In financial terms, a one percent advantage means a player will win $10 per hand, on average, betting $1,000 per hand. If the player is dealt 50 hands per hour, this translates into $500 per hour profit. However, since play and luck vary, one needs a tidy bankroll to stay in the game. One rule of thumb suggests 400 times the average zero count bet. This often amounts to tens, if not hundreds of thousands of dollars for professional gamblers.
People love games of chance, and have been playing them for thousands of years. The desire to win at such games was the motivation behind the development of probability theory, and my treatment of this subject would certainly be incomplete without paying appropriate homage to humankind's search for an edge.
For more information, the Wikipedia entry on card counting provides a good overview as well as useful links to other resources.
And good luck---it always helps. And, by the way, the movie-makers probably made a lot more money in the end than the card counters.
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