M: The next time you go Playground, Do not participate in "luck"Game! A lot of people play this game are fools, because they thought they would not mistakes.

M: "luck" is filled with three games in a cage Dice Flip shake dice roll cage to make people play from 1-6 can bet any number, as long as a number of dice appear, he said when he got the money he bet several participants often think: If this cage There is only one dice, I bet the number can only appear once in six times. If there are two dice, the six will appear twice. There are three dice, they will have to win six times, this It is like Gamblingʱ??

M: "But, my chance even better if I bet on a number, such as 5, bet a dollar if there are two dice, then 5, I won two dollars; if the three dice are 5!. I won three. This game is certainly good for me! "

M: Because patrons think so, no wonder Casino The operator will become a millionaire! Can you explain why "luck" game will win large sums bet casino master it?

"Luck" is playing in the United States and many overseas gambling casino play. In the UK, this gambling can be traced back to the early nineteenth century, it was known as the "Hanjin." Recently called "the cage." In the UK and Australian pub, this three dice on gambling and India is a spade, a box with a red heart, a plum - a crown, an anchor, and called the crown and anchor.

In the playground, the operator is to attract customers and shouted: "Every three wins, three people lost!" This gives a strong impression that it is fair.But if the three dice show the numbers are not the same every time, then this is indeed a fair bet play.In a cage after each shake, the operator can win three hands losers from three dollars (assuming each bet a dollar), the three winners paid three dollars.However, the operator fortunately, often showing the same number in two or three dice.If there are two dice is the same number, so he incorporated into the four dollars. Pay three lines, earn a dollar.If there are three dice are the same number, then he would receive five dollars, to pay three dollars, earn two dollars.It is these double and triple the number to make the number of casino owners earn a lot of money.

Using formulas to calculate the ratio of primary to win the casino is a tricky job. Ordinary student is best to all the 216 kinds of possible cases falling List All three dice. At this time, they will find only 120 cases, of which three a dice different, 90 kinds are the same two points, six kinds are three points are the same [*] It is assumed that this gambling drama played 216 times, resulting in all 216 kinds of results. Every game, six people for six different numbers for each bet a dollar. casino gambling Lord in 216 collected 216 * 6 = 1296 yuan [?].

When three dice are not at the same time, he had to pay 6 dollars (three winners per two dollars), a total of 120 such cases, so he paid 6 * 120 = 720 dollars. When two dice the same (total 90 cases), he shall pay a point person 2 * 90 = 180 dollars. There are two different points paid people 3 * 90 = 270 dollars. When all three dice When a number of points (total six cases), he shall pay 6 * 4 = 24 dollars. In this way, he paid a total of 1194 dollars, net profit of 102 yuan.

Divided by 1266 yuan to 102 yuan, come to the casino main interest rate of 7.8 +%. This means that he can expect after a long period of time gambling, gamblers for each dollar bet, he will be 7.8 points more than that. A press any number gambler gambling, 216 cases, only 91 kinds of circumstances in which he appeared at least once this number [?], so he won the probability of a dollar is 91/216 than 1 / 2 is much smaller.

[*] This result can be calculated using the formula permutations and combinations of three dice different, it can be seen as three dice are arranged to take 1-6 six digits: A63 = 6 * 5 * 4 = 120 three dice. Two points, as can be seen in three dice arrangement 1-6 take two numbers, two points is a single dice can take turns three dice in a total of three, so this number is 3 * A62 = 3 * 6 * 5 = 90 three dice with only 1-6 full six kinds, for a total of 216 kinds of situations is another algorithm: a combination of the three dice each may take 1-6 six numbers is 6 * 6 * 6 = 216-- Annotation

[?] When playing gambling, gambling to come up with a dollar per person. If he wins, we get back two dollars, if he lost, to lose this money .-- Annotation

[?] This can be calculated as follows: When he chooses a number, there are three win situation: first, three dice are his chosen number, then only one possible second and third in two. Dice is the number he chose this time to take another dice to any one of the other five numbers, but numbers that separate the three dice turn three, so the number is 3 * 5 = 15. Third, the only one of the dice is he To the number, this elbow the other two dice may be any of a number of the remaining 5 5 * 5 = 25. But each one of the three dice desirable number he wanted, so a total of 3 * 25 = 75 kinds of above three cases a total of 75 + 15 + 1 = 91 kinds.