Probability Applied in everyday life! By Praneetha Mukhatira 03/15/2004

1. Probability Applied in everyday life!By Praneetha Mukhatira03/15/2004

2. How Safe is it to Fly? • “NTSB studies show that from 99-02,scheduled US carriers averaged only 0.2 fatal accidents/10,000 flight hours less than half the fatal accident rate for the 4 yr period a decade earlier.” • Is a 5 hr flight the same as 5 1 hr flights? • Mortality risk measure: If a person chooses a flight at random; probability of being killed during the flight = ? Period Death risk/flight 1987-1997 1 in 7 million 1997-2003 1 in 7 million At this level of risk, she would on an average go for 19,000 years before succumbing to a fatal accident.

3. Would you rather drive? • Motor vehicle accidents cost about 40,000 lives and 1.5 million injuries each year. • Fatality rate in city traffic is more than double that of freeways. • Fatality decreases by half when seat belts are worn. • Death risk per 100 miles for a safe driver is 1 in 10 million. • For every hour you save by traveling by jet rather than by car, there is a bonus :67 seconds increase in life expectancy. • There are approximately 40,000 auto fatalities annually in this country, so in any given three-week period, there would be about 2,300 fatalities. The area around Washington has a population of about four million, or 4/280 of the population of the U.S., so as a first approximation, we could reasonably guess that 4/280 times 2,300, or about 30 auto fatalities, would occur there during any three-week period. Attention must then be paid to the ways in which this area and its accident rate are atypical

4. No Smoking Please!! • Cigarette smokers include 40% of American adults and 85% of lung cancer sufferers. • Let Q=number of Americans who get lung cancer per year. N=American adults. If 40% Americans smoke get 85%of lung cancer cases, their annual lung cancer rate is: 0.85Q/0.4N = 2.25(Q/N).For Non-smokers, it is: 0.15Q/0.6N =0.25(Q/N). • Cigarette smokers have 9 times annual lung cancer than passive smokers. • Approx. 8%of all US deaths each year caused by Lung cancer. If V=lung cancer death risk for non-smokers, 0.6V+9(0.4V) =0.08. V=0.019,For Smokers = 9V =0.17. • Passive smokers fall into the same state of impaired performance as the light smokers.

5. Safe Massachusetts • Total population of Boston City ~6 million. • Number of Homicides =143 in 1990,39 in 2002.i.e a drop from 0.02% to 0.006% • Mass population is 6175000. Total crimes =201460 Crime rate = 3% • N.Y. city population = 7.5 million. Killings in 2002=2200.Homicide risk =1 in 3400.Over a life span of 70 yrs, cumulative murder risk for a citizen is 70*(1 /3400) =1 in 49.

6. Lotteries • House percentage is high. • Psychological factors: easy to buy, many ways to win, rules keep changing. • Ex:5 numbers out of 90;40% as house percentage. Chances of getting all 5 is 5/90*4/89*3/88*2/87*1/86 ~1 in 40 million. Getting 4 numbers is 1 in 100,000. • Strategy: Player picks unpopular numbers b’coz it gives bigger payoff if you do win. It included numbers at the edge and corners; they were picked 0.5 times lesser than the average. Hence if that number was picked, he had an expectation at least twice as large as the average 60 cents on the dollar. • In Mass. “The Big game Mega Millions” on Friday Feb.20th,2004, Pick 5 out of 52 and 1 out of 52 . The probability was as follows: 5 plus Mega Ball= 1 : 135,145,920 5 match ONLY =1 : 2,649,920, 4 plus Mega Ball=1 : 575,089, Mega Ball ONLY=1:52.

7. Keno • House percentage is high. Payoff is fixed in advance. • Player picks 20 numbers out of a total of 80 numbers. The house draws 20 numbers and player wins if enough of his numbers are the same. • Ex: Popular 8 ticket costing $6.50,If 5 match you win$5, if 6 then $50,7 then $1100,8 then $12,500.The probability that all 8 will be drawn is 20/80*19/79… = 1/230115. • Your best bet would be to buy a 5 spot card ,which has a probability of 0.019 with payoff of $5. • One version is called "Top and Bottom Keno". In this version you don't pick any numbers, instead you write a "T" on the top portion, or a "B" on the bottom of the ticket. In each portion there are twenty numbers. If thirteen or more numbers in either portion are drawn out of the bowl you win.

8. Casino1-Slot Machines • Big House edge • Study at Wisconsin Ex: There are 3 dials with 20 positions on each dial. Total of 20* 20 * 20 = 8000 different combinations at which dials could stop. Maximum payoff was 62 units for 3 bars. There was 1 bar each on dial1 & 3 and 3 on dial 2. 3 ways to win out of 8000 possibilities.9 other ways to win smaller amounts. It gives you an expectation of $0.223 on the dollar. • When all ways of winning at any slot machine are added, you are lucky if you get $.75 for each dollar dropped in.

9. Casino2-Roulette • Betting on the spin of a wheel is an old form of gambling. Combining a ball to it is credited to Pascal. • In the European version, there are 37 pockets,0-36.Players bet on nos. from 0-36 and are paid off at odds. On an average, the house breaks even when ball falls on any of the 36 numbers and collects 100%when it falls on 0.House % is 1/37~2.7%. • American Casinos have double zeros in the zero pocket. House %=2/38~5.56% • 36 is divisible by 2,3,4,6,12,18,You can bet on combinations of these nos. with payoffs of 17 to 1,11,8,5,2 1 .But the wheel has 38 numbers, hence house % is same. • The house % is largest when you make 5 letter combination. It pays 6 to 1,house edge is 7.89%.

10. 1.) Straight Bet / The chip can be put anywhere on the layout on one of the 38 numbers. It has to be completely in the square surrounding the number you choose because if it is not the dealer could mistake it for a different bet. This bet offers the highest odds in this casino game. You get paid a 35:1 odd if the ball lands on your number. 2.) Split or Two number Bet / In a split bet you place your chip on the line between two neighboring numbers. In this case the numbers are 12 and 15. You win if the ball lands on either of those numbers. The odds on this bet are 17:1. 3.) Street or Three number Bet / This bet allows you to cover an entire row of the table. You make this bet by placing your chip on the outside line of the row you want to bet on. You win if one of those three numbers comes up. The odds here are 11:1. 4.) Corner Bet / This bet covers four numbers. To make it, you have to put your chip right in the middle of the four numbers where they join corners. In this case they are 20, 21, 23 and 24. The odds on this one are 8:1. 5.) Five Number Bet / Here you have to place your chip in the only possible 5 number street available. This bet only covers the numbers 0, 00 1, 2 and 3. If one of these numbers comes up you will get paid 6:1 odds. 6.) Six Number Bet / This bet makes it possible to cover two rows of three numbers each. You have to place your chip on the outside line of the two rows you want to cover. In this case the numbers are 28, 29, 30, 31, 32 and 33. The odds on this particular bet are 5:1. 7.) Any Red or Black Bet / In this case you put your chip either on the black or the red field on the outside. This covers all the black or red numbers on the field. The odds are 1:1. 8.) Any Low or High Number Bet / This bet divides the field of numbers into two groups. The numbers from 1 to 18 (low) and from 19 to 36 (high). You bet on whether the next number that comes up is between 1 and 18 or 19 and 36. In either case if 0 or 00 shows up you lose. The odds are 1:1 also. 9.) Any Even or Odd Bet / Here you bet if the next number that comes up is either even or odd. Same as the bet above each of those two fields covers 18 numbers. 0 or 00 is not considered either even or odd, so if it comes up you lose. The odds are 1:1 on this bet as well. 10.) Dozen Bet / This divides the numbers into three dozen. Each one of them covers 12 numbers (1 to 12, 13 to 24 and 25 to 36). As shown above your bet would be on the first dozen, this means all the numbers from one to twelve would be covered. In case you win you get paid 2:1. 11.) Column Bets / There are three columns, namely 1st, 2nd & 3rd. Either of them covers 12 numbers. In the example above the bet would be on the 2nd column. If any of the numbers included in the column show you win 2:1.

11. Casino3-Craps • Craps is a game played with two dices and up to eight players participating. The game starts with what is called a "Come Out" roll made by the so called "Shooter". This is the player currently rolling the dices. The shooter wins if he rolls a so called "Natural" which is a 7 or 11, and loses if the roll is a 2, 3 or a 12. This Is called "Craps". Rolling any of the remaining numbers (4,5,6,8,9, or 10) is known as the Point. If the shooter establishes a point at the come out roll, he has to roll another point and then a 7 to win the game. Rolling a seven right after the first point would mean he loses and the dices and they go on to the next player. In case you are the next shooter and you don't want to roll the dices you always have the option to give them right to the next player without rolling yourself.

12. 1.) Pass Line Bet / This is the most popular and simplest bet in craps. You bet that the shooter wins his game. This bet can be made at any time but is generally made before the come out roll. The odds are 1:1 on this bets. Winning Possibility 49.30%,H.P.=1.41%2.) Don't Pass Bet / This is exactly the opposite of the above. You bet the shooter does not win his game. Odds are 1:1 too. Winning Possibility 50.71%,H.P.=1.36%3.) Odds Bet / When the shooter establishes the point in the come out role you can place an odds bet as an option to your Pass Line or don't Pass wager. In this bet the casino has absolutely no advantage because you don't bet against it. Basically you strengthen you pass or don't pass wager. A winning odds bet pays you the true odds which are 6:5 for a 6 and 8 roll, 3:2 for a 5 and 9 roll and 2:1 for a 4 and 10. Winning Possibility 33.30%,H.P.=0.51%4.) Come Bet / The come bet works exactly like the Pass line bet but you make the bet after the point is established. The next roll becomes the come out roll for your bet. A come bet wins with 7 and 11 and loses with 2, 3 and 12. All other rolled numbers cause your wager to be moved to that particular number (the x is the spot where your wager is moved to, in this case the result of the roll was an 8). For you to win, the point has to be re-established before a 7 is rolled. The odds are 1:1 also. Winning Possibility 49.30%,H.P.=1.41%5.) Don't Come Bet / The opposite of the explained above. Odds also are 1:1.Winning Possibility 50.71%,H.P.=1.36%6.) Field Bet / You bet the outcome of the next roll will be a 2, 3, 4, 9, 10, 11 or 12. If the dices show 5, 6, 7 or 8 you lose. The odds are 2:1 on a 2 and 12 roll and 1:1 on a 4, 9, 10 and 11. Winning Possibility 44.44%,H.P.=5.56%7.) Place Bet / Here you bet that a certain number will be rolled before a 7. The odds are 7:6 on a 6 and 8 roll, 7:5 on a 5 and 9 roll and 9:5 on a 4 and 10.Winning Possibility 45.45%,H.P.=1.52%8.) Buy Bet / A buy bet is the same as the place bet but it pays the true odds with a 5% charge with every win. Winning Possibility 45.30%,H.P.=2.73%9.) Big 6 and 8 / Here you bet that a 6 or a 8 is rolled before the next 7. Odds are 1:1. Winning Possibility 45%,H.P.=9.09%10.) Any Craps / You bet the next roll will be a 2, 3 or 12. The odds on this one are 7:1.Winning Possibility 11%,H.P.=11%11.) Hardways / This is a place bet on one of the doubles, 2+2, 3+3, 4+4 and 5+5. The odds are 7:1 on a hard 4 and hard 10 (2+2 and 5+5) and 9:1 on a hard 6 and a hard 8 (3+3 and 4+4). Winning Possibility 9.9%/11.11%,H.P.=9.9%,11.11%%

13. Casino4-BlackJack • The game starts with every player making their opening bets. After all the players placed their bets the dealer will start dealing the cards. Starting with the player to his left he gives every player one card, face down, including himself. This is the dealers down card. Then he deals a second round of cards, face down but this time the card he deals himself will be face up. This is the dealers up card. You now can look at both of your cards and find your total by simply adding the values of your cards. • The values of the cards in Black Jack from two to ten are at face value. Jacks, Queens and Kings count ten and the Ace counts eleven or one. The Ace always counts eleven except if your total exceeds 21 - then the value of the Ace is reduced to one. A hand with one Ace having the value of eleven is called a soft hand and a hand with all Aces having the value of one is called a hard hand. In Black Jack for instance, if you get an 8 and Ace dealt it would be a soft 19 while an 8, 10 and Ace would be a hard 19. Getting a start total of 21 is called a Black Jack and you have to show your hand immediately. If the dealer's up card is an Ace he checks for a dealer black jack first and then continues the game. Exceeding a total of 21, and already counting all the aces you have in your hand as one, means you are bust and lose your bet.

14. By turn each player will then have to make one of the following five decisions. • Hit / If you are not satisfied with your current total you can ask the dealer to hit you which means he deals you another card in addition to your two. You are hit until you are satisfied with your total, or until you bust. • Stand / You stand if you don't want any more cards. • Double / If you think you will win without getting more than one card you double. You have to add an amount equal to your original wager and receive only one card. If your total is higher than the dealer's after receiving the card you win. • Split / If your starting hand contains two cards of the same type (i.e. two 9's) you can split them up into two new hands. You have to add an equal amount to your wager and get two more cards dealt forming two separate new starting hands. • Insurance / Insurance is offered to the players if the dealer's up card is an Ace, to protect against a dealers Black Jack. You will have to pay half of your original bet and will get 2:1 odds when the dealer has a Black Jack. Unless you also have a black jack your original bet is lost. • Surrender / This decision is quite rare and not offered is most casinos. After you see your starting hand and the dealers up card and you don't think you can win, you have to give your cards back to the dealer immediately. If you surrender you will only lose half of your original bet. You cannot surrender if the dealer has a Black Jack. • After all the players have made their decision the dealer will then play his hand. The playing of the dealer's hand must follow certain rules. He must hit on every total less than 17 or otherwise stand. Some casinos even let the dealer hit when he has a soft 17. The rules which the dealer has to follow will be written clearly on the Black Jack table, so there will be no confusion. • You win if either the dealer busts or has a total less than yours. The odds are 1:1. If the total is the same it's a draw or a push and your original wager is returned to you. A black jack beats an ordinary 21 and is paid 3:2 odds

15. Baseball • Too Many Seven-Game Series? . Among the best-of-seven World Series (first team to win 4 games wins) of the past 50 years, have there been more that went the full seven games than probability theory would have predicted? • In the period from 1952 to 2002, 24 of the 50 Series (or 48 percent) went the full seven games and the likelihood of this many or more 7-game Series is a small and statistically significant 1 percent. Most of these Series occurred in the period 1952 to 1977. If, however, the analysis is extended back to include all World Series, 35 of the 94 (or 37 percent) went the full seven games, higher than expected, but not statistically significant. • If we pitted the American League champion ,say, Boston Red Sox against, say, the National League All-Star Team , the All-Stars might very well be favored against the Red Sox, even if the Red Sox were favored against every particular NL team

16. Assume that teams, A and B, are equally matched so that the probability of each winning is 1/2 or 50 percent. The probability of team A winning four consecutive games is (1/2)^4 or 1/16. By assumption, this is also the probability that team B will win four consecutive games, so the probability that the series will go four games is 1/16 plus 1/16 or 1/8 — 12.5 percent. The probability that team A wins in five games is the probability that the sequence of game wins will be one of four possibilities: BAAAA, ABAAA, AABAA, or AAABA. Each of these possibilities has probability (1/2)^5 or 1/32. Thus the probability that one of these four sequences occurs and team A wins the Series in five games is 4/32. By assumption, this is also the probability that team B will win the Series in five games, so the probability that the series goes five games is 4/32 plus 4/32 or 1/4 — 25 percent. • We can determine the probability the Series lasts six or seven games in a different way. To last six or seven games, the Series must have one team ahead three games to two at the end of five games. If the team that's ahead wins, the Series ends after six games, whereas if the team that's behind wins, the Series goes seven games. The teams are assumed to be evenly matched, so the probability that the Series goes six games equals the probability that it goes seven games. • Since the probability that the Series ends in four or five games is 37.5 percent (the sum of 12.5 percent and 25 percent), the probability that it ends in six or seven games is what's left over or 62.5 percent (100 percent minus 37.5 percent). Dividing 62.5 percent in half, we get that the probability the Series lasts six games is 31.25 percent, the same as the probability that it lasts seven games.

17. Sad dam(n) lie! • Eleven million people went to the polls in Iraq when Saddam was around, and, the Iraqi news media assured us, 100 percent of them voted for Saddam Hussein for president. Let's just for a moment take this vote seriously and assume that Hussein was so wildly popular that 99 percent of his countrymen were sure to vote for him and that only 1 percent of the voters were undecided. Let's also assume that these latter people were equally likely to vote for or against him. Even given the absurdly generous assumptions above, there would be 110,000 undecided voters (1 percent of 11 million). The probability of a 100 percent vote is thus equal to the probability of flipping a fair coin 110,000 times and having heads come up each and every time! The probability of this is 2 to the power of minus 110,000, or a 1 preceded by more than 30,000 0's and a decimal point. This would be the cosmic mother of all coincidences!

18. R U the Sniper? • Early in the sniper case the police arrested a man who owned a white van, a number of rifles, and a manual for snipers. It was thought at the time that there was one sniper and that he owned all these items, so for the purpose of this question let's assume that this turned out to be true. • Given this and other reasonable assumptions, which is higher — a.) the probability that an innocent man would own all these items or b.) the probability that a man who owned all these items would be innocent? • The second probability would be vastly higher. To see this, let us make up some illustrative numbers. There are about four million innocent people in the area and, we'll assume, one guilty one. Let's estimate that 10 people (including the guilty one) own all the three of the items mentioned above. The first probability — that an innocent man owns all these items — would be 9/4,000,000 or less than 1 in 400,000. The second probability — that a man owning all three of these items is innocent — would be 9/10. Whatever the actual numbers, these probabilities usually differ substantially. Confusing them is dangerous (to defendants).

19. Picking the wrong guy • The error rate is alarming among eyewitnesses to crimes. They have discovered a number of factors that significantly influence the likelihood that witnesses will correctly pick the culprit out of a lineup . Despite the fact that eyewitnesses are usually quite certain of who and what they've seen, the probability of a correct identification (after people have seen videotape of a simulated crime, for example) is frequently as low as 60 percent, and, what's worse, innocents in the lineup are picked up to 20 percent or more of the time • Picking the Biased Penny: Assume that you have three suspect pennies lined up before you. You're told that one of these pennies, the culprit, lands heads 75 percent of the time, and that the other two, the innocent suspects, are fair coins. You know nothing else about the pennies, but you did previously observe that one of the coins was flipped three times and landed heads all three times. Having witnessed this and realizing that the biased penny is much more likely to behave in this way, you identify this coin as the culprit. How likely are you to be right? • The calculations are formally analogous to what we do when we change our estimate of the probability of a suspect's guilt after the testimony of an eyewitness. Identifying a biased coin on the basis of the evidence of three consecutive heads is mathematically the same as identifying a human culprit on the basis of an eyewitness' memory. Let us use Bayes’ theorem.

20. First let's determine how often we will see three consecutive heads if one of the three pennies is chosen at random and flipped three times. • One-third of the time the culprit coin will be chosen and, when it is, heads will come up three times in a row with probability 27/64 (3/4 x 3/4 x 3/4), and so 14.1 percent of the time (.141 = 1/3 x 27/64) the culprit will be chosen and will land heads three times in a row. • Two thirds of the time a fair coin will be chosen and, when it is, heads will come up three times in a row with probability 1/8 (1/2 x 1/2 x 1/2), and so 8.3 percent of the time (.083 = 2/3 x 1/8) a fair coin will be chosen and will land heads three times in a row. • The coin selected will thus land heads three times in a row 22.4 percent of the time (14.1 percent + 8.3 percent). • Of the 22.4 percent of the instances where this happens, most occur when the coin is the culprit; specifically [14.1 percent / (14.1 percent + 8.3 percent)] or 63 percent of them do. That is, we will be right 63 percent of the time if we identify a coin that's landed three times in a row as the culprit among three pennies. (Of course, the penny may cop a plea by pleading insanity and admitting to being unbalanced.) • The difference between the probability the penny is the culprit given that it's landed heads three times (63 percent) and the initial probability the penny is the culprit (33 percent). The information gain is thus 30 percent.

21. A Puzzle • Here's the situation. Three people enter a room sequentially and a red or a blue hat is placed on each of their heads depending upon whether a coin lands heads or tails. • Once in the room, they can see the hat color of each of the other two people but not their own hat color. They can't communicate with each other in any way, but each has the option of guessing the color of his or her own hat. If at least one person guesses right and no one guesses wrong, they'll each win a million dollars. If no one guesses correctly or at least one person guesses wrong, they win nothing. • The three people are allowed to confer about a possible strategy before entering the room, however. They may decide, for example, that only one designated person will guess his own hat color and the other two will remain silent, a strategy that will result in a 50 percent chance of winning the money. Can they come up with a strategy that works more frequently? • Most observers think that this is impossible because the hat colors are independent of each other and none of the three people can learn anything about his or her hat color by looking at the hat colors of the others. Any guess is as likely to be wrong as right.

22. The strategy that enables the group to win 75 percent of the time: It requires each one of the three to inspect the hat colors of the other two and, if the colors are the same, then to guess his or her own hat to be the opposite color. When any one of the three sees that the hat colors of the other two differ, then he or she must remain silent and not make a guess. • The eight possibilities for the hat colors of the three people are RRR,RRB, RBR, BRR, BBR, BRB, RBB, and BBB. In six of the eight possibilities (in bold), exactly two of the three people have the same color hat. In these six cases, both of these people would remain silent (why?), but the remaining person, seeing the same hat color on the other two, would guess the opposite color for his or her own hat and be right. In two of the eight possibilities (in italics), all three have the same color hat and so each of the three would guess that their hats were the opposite color and all three of them would be wrong. • If one were to run this game repeatedly, the number of right and wrong guesses would be equal even though the group as a whole would win the money six out of eight times or 75 percent of the time. That is, half of all individual guesses are wrong, but three-fourths of the group responses are right!