{"id":404,"date":"2011-08-21T10:33:03","date_gmt":"2011-08-21T02:33:03","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=404"},"modified":"2024-02-16T21:13:34","modified_gmt":"2024-02-16T13:13:34","slug":"dice-and-polynomials-part-1","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2011\/08\/dice-and-polynomials-part-1\/","title":{"rendered":"Dice and Polynomials – Part 1"},"content":{"rendered":"
\"Can<\/a>
Can polynomials help calculate probabilities?<\/figcaption><\/figure>\n

Imagine you have two coins. One side is blank, and one has a single dot. You flip the coins. How many ways can you get 0 dots? How many ways can you get 1 dot? How many ways can you get 2 dots?<\/p>\n

If you got answers like “1 way \/ 2 ways \/ 1 way” you got it right.<\/p>\n

Now, another question. If I multiply together 1.x0<\/sup> + 1.x1<\/sup> and 1.x0<\/sup> + 1.x1<\/sup>, that is, (1+x) times (1+x), what is the coefficient of x0<\/sup> ? of x1 <\/sup>? of x2<\/sup> ? If you got answers like “1 \/ 2 \/ 1”, you got it right.<\/p>\n

If you ask similar questions for three coins, or (1+x)3<\/sup>, you still get the same answers : 1, 3, 3 and 1. There’s a good reason for this – the question “how many ways can you get 2 dots” is, in a way, the same question as “what is the coefficient of x2<\/sup> ?”. This has huge implications, it gives a great new way to calculate probabilities, and more<\/a>.<\/p>\n

Imagine I have a coin. Or a die. Whatever. Some of the faces have dots. I’ll represent each dot by ‘x’. The dots showing up on one side will be all multiplied together, so a face with three dots would be x3<\/sup>. A face with one dot would be x3<\/sup>, or x, and one with no dots would be x0<\/sup>, or 1. So the two sides of my coin are 1 and x, and the 6 sides of a normal die are x,\u00a0x2<\/sup>, x3<\/sup>, x4<\/sup>, x5<\/sup> and x6<\/sup>.<\/p>\n

Now, what about the whole coin or die? To represent the whole die, I’ll add up all its faces. The coins at the start are each 1+x, and a normal die is x + x2<\/sup> + x3<\/sup> + x4<\/sup> + x5<\/sup> + x6<\/sup>. Or, if I had a strange die whose faces were 1, 2, 2, 2, 3 and 3, it would be represented by the polynomial x + x2<\/sup> + x2<\/sup> + x2<\/sup> + x3<\/sup> + x3<\/sup>, that is, x + 3x2<\/sup> + 2x3<\/sup>.<\/p>\n

In these polynomials, the coefficient of each power of x represents the number of ways to get that number of dots. So, for my strange die, the coefficient of x2<\/sup> is 3 because there are three ways to get a 2.<\/p>\n

Now, why are we going to all the trouble to write down polynomials representing these dice? The comes when you want to toss two or more of these dice, and count how many ways there are to get certain numbers. If I<\/p>\n