Probability (1970)

[Music] [Music] years ago in the land of howan there lived a a Raja R although he was Raja of all the land and the richest man for miles around he was terribly bored what a drag you win again S I know that's the trouble yes that was the trouble for the Raja dearly loved to play games but he was so good at them that nobody in all of howan had ever beaten him it didn't seem to matter what game he played maraja never lost and of course it's no fun to win all the time I've got to find someone who can beat me at something so the Raja began importing famous gamesters from far and near to try their skill against him Royal Flush you lose again I've got to find someone who can beat me at something I know I'll issue a proclamation here ye here ye the most excellent Raja haja of how stand hereby decrees that someone come forward and defeat him in fair play at any game whatever if he does he shall be rewarded with diamonds and rubies [Music] ah if on the other hand no one is able to defeat the Raja by sundown tonight everybody's taxes will be doubled why don't you ask Ali cart he'd know if anyone would where can I find this Ali C in the street of a thousand Knights excuse me are you Ali C yes but don't bother me I'm upset about my nephew Ali mod why all he does is play games all day where can I find this nephew of yours over in the alley playing some game no doubt EX excuse me are you Ali mod what oh yes that's me what are you doing studying a new game I just made up are you uh very good at games there's no one in the whole land that can beat me ah Come with Me sire sire here is someone who thinks he can beat you wonderful what game chess Checkers part Jayy oh no your highness I'm sure no one can beat you at those games but I can beat you at a game I made up myself a game you made up yourself eh all right but no cheating my guards will be watching oh I'll play fair your highness we'll need a Blackboard and some chalk a Blackboard and some chalk all right your highness let me show you how this game goes first I'll write each of our names names on the board next each one of us chooses one of these numbers then I spin the spinner if the arrow points to the number you choose you get a point if it points to my number I get a point if it points to a number that is neither yours nor mine no one gets a point we keep track of the points over here and we keep playing until we can both see that one of us is getting further and further ahead I see and you think you can beat me at this game let's say that I don't think you can beat me all right let's try it I can't see any reason why one number should come up any more often than any other number so I think this game will be a tie no matter what number I choose I'll choose one and I'll choose four that's one for you no one gets a point for that one ahuh that's one for me well Ali it looks like a draw to me neither one one of us seems to be getting very far ahead of the other and just as I said before we began playing there is no reason that any of these numbers should come up any more often than any of the others so in the long run we should each get the same amount of points Ali I didn't beat you but you didn't beat me either and that was in my Proclamation have you got another game yes I do your highness and I think I can defeat you I hope he doesn't in a hurry it's getting close to sundown this game we play with two spinners this time we get a number by spinning both the Spinners and adding the two numbers with arrows pointing to them for instance these two spins give us six everything else about the game is the same we each pick a number whenever a person's number comes up he gets a point and we play until we can both see that one of us is getting further and further ahead now let's see the smallest number that could come up would be two and the largest would be eight it looks to me like any number from 2 to 8 could appear that's correct your highness well I still don't see why any number should come up any more often than any other number I think it's going to be a draw again I think I can win your h only one thing what's that you picked first in the last game let me pick first this time very well I don't see what difference it makes who picks first I pick five I'll pick seven h no one gets a point for that one aha a point for me well your highness all right Ali I admit you are getting further and further ahead you win the Raj [Music] lost I lost but I don't understand why why did five come up more often than seven in fact five came up more often than any other number why I'll explain your highness you were right when you said that any of these numbers would come up about as often as any of the other numbers but let's take a look at the combinations we can get we can get one and one or one and two or one and three and so on as you can see there are 16 possible combinations each is as likely to come up as any other now let's imagine that we stack these combinations in piles that add up to the same number so the one and one combination would go in the two pile the one and two combination would go in the three pile and so would the two and one combination and so on so you see your highness even though any one of these combinations will come up about as often as any other there are four combinations that add up to five and only two combinations that add up to seven and so five will come up about twice as often as seven in fact five will come up more often than any other number well that's all well and good and I can see that the chances were very much in your favor but isn't it possible that I could have been very lucky and won anyway but I knew the odds were in my favor and that I'd probably win so I took the chance thank you Ali I feel much better and here is your reward ooh ah thank you your highness don't just stand there get me the checkers well nobody likes to lose all the time Ali who would have won the game if you had each chosen two numbers say if one of you had chosen three and four the other five and six you can figure that out in the same way that I figured out that five would occur more often than seven and once you figured that out see if you can divide the numbers into two sets so that each set has an equal chance of occurring [Music] n [Music]