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

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

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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 Qnumber of Americans who get lung cancer per
  year. NAmerican adults. If 40 Americans smoke
  get 85of 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. 8of all US deaths each year caused by
  Lung cancer. If Vlung cancer death risk for
  non-smokers, 0.6V9(0.4V) 0.08.
• V0.019,For Smokers 9V 0.17.
• Passive smokers fall into the same state of
  impaired performance as the light smokers.

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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 20022200.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.

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Lotteries

• House percentage is high.
• Psychological factors easy to buy, many ways to
  win, rules keep changing.
• Ex5 numbers out of 9040 as house percentage.
  Chances of getting all 5 is 5/904/893/882/871/
  86 1 in 40 million. Getting 4 numbers is 1 in
  100,000.
• Strategy Player picks unpopular numbers bcoz 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
  Ball1 575,089,
• Mega Ball ONLY152.

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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
  win5, if 6 then 50,7 then 1100,8 then
  12,500.The probability that all 8 will be drawn
  is 20/8019/79 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.

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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.

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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 100when it falls on 0.House
  is 1/372.7.
• American Casinos have double zeros in the zero
  pocket. House 2/385.56
• 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.

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• 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 351 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 171.
  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
  111. 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 81. 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 61 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 51. 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 11.
  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 11 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 11 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
  21. 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 21.

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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.

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• 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 11 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
  11 too. Winning Possibility 50.71,H.P.1.363.)
  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 65 for a 6 and 8 roll, 32 for a 5 and
  9 roll and 21 for a 4 and 10. Winning
  Possibility 33.30,H.P.0.514.) 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 11 also.
  Winning Possibility 49.30,H.P.1.415.) Don't
  Come Bet / The opposite of the explained above.
  Odds also are 11.Winning Possibility
  50.71,H.P.1.366.) 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 21 on a 2 and 12 roll and 11
  on a 4, 9, 10 and 11. Winning Possibility
  44.44,H.P.5.567.) Place Bet / Here you bet
  that a certain number will be rolled before a 7.
  The odds are 76 on a 6 and 8 roll, 75 on a 5
  and 9 roll and 95 on a 4 and 10.Winning
  Possibility 45.45,H.P.1.528.) 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.2.739.) Big 6
  and 8 / Here you bet that a 6 or a 8 is rolled
  before the next 7. Odds are 11. Winning
  Possibility 45,H.P.9.0910.) Any Craps / You
  bet the next roll will be a 2, 3 or 12. The odds
  on this one are 71.Winning Possibility
  11,H.P.1111.) Hardways / This is a place bet
  on one of the doubles, 22, 33, 44 and 55. The
  odds are 71 on a hard 4 and hard 10 (22 and
  55) and 91 on a hard 6 and a hard 8 (33 and
  44). Winning Possibility 9.9/11.11,H.P.9.9,11
  .11

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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.

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• 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 21
  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 11. 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 32 odds

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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

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• 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.

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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!

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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).

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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!