Probability-Tools

=**Probability Tools**=

A B C Then, you would put the two other possible candidates for next spot next to each A-B C
 * Counting Tree-** A diagram that shows all the possible outcomes for a scenario. A manual version of Permutations. You make one by making a first column of all the possibilities for the first choice. For example, if you are trying to solve a problem of order, such as what order people could get into a car into. Suppose there are 3 people, labeled A-C. You would start a counting tree with three columns, A, B, and C.

B-A C

C-A B

You would keep doing this until you have the full tree.

A-B-C C-B

B-C-A A-C

C-A-B B-A

You then have a counting tree. To find the number of possible orders, you simply count the number of branches that have all three letters in them. In this case, paths.


 * Pascal’s Triangle-**A tool used very commonly in probability. You create one by making a triangle with numbers. The top tip of the triangle is a one. You then work your way down, connecting the numbers with “branches” with all the lower numbers being the sum of the branches above them. All the blank spaces above or next to rows and columns are considered to be zeros.

/ \ 1 1 / \ / \ 1 2 1 / \ / \ / \ 1 3 3 1 / \ / \ / \ / \ 1 4 6 4 1 / \ / \ / \ / \ / \ 1 5 10 10 5 1

This triangle can be used to find the numerator in many probability problems, such as the coin problem. If you had to find the probability of getting 3 heads in 7 flips, you would use 7 permutation to find that there are 128 different possible outcomes possible, the denominator. Saying that X is the number of flips, and Y is the number of heads, you would go to the Xth row of the triangle (The top vertex does not count), and the Yth number in that row. (This does not count the first and last one in the row) This number would be the numerator in your fraction. With 3 heads in 7 flips, you would go to the 7th row, and the 3rd number in that row. This number is 35, the numerator for the fraction.


 * Permutation-**Permutations is a way to quickly find out all of the possible outcomes for a scenario, written with an exclamation mark at the end. There are two types of permutations, the ones where you are allowed to repeat digits, and you are not. To find the permutation of a number is simple, yet varies per type. If you wish to find the permutation of a number where you can repeat digits, you simply square the number. If you are using it in a particular probability experiment, you may need to vary this. For example, in our problem of 3 heads in 7 flips, you will need to multiply 2x2x2x2x2x2x2, as you need to find all the possible variations with 7 flips, with two possibilities per flip. If you wish to find a permutation where you are not allowed to repeat digits, you keep multiplying your number by the next lowest number until you get to zero. For example, finding 7! For this type of permutation would look like 7x6x5x4x3x2x1. For example, if you wished to find how many ways there were for five different colored cubes to be arranged, you would have to find 5!, meaning 5x4x3x2x1. (In this case 120) Note: In permutations, 0=1.


 * Standard Deviation-**Measures how far away a typical occurance will be from the average. This means that if the average chance of rolling a one on a die is 1/6, the standard deviation will measure the range of where the actual experiment will fall in. The formula for finding Standard deviation is SD=Square Root of T x O x N. T=Trials, O=Chance of occurrence, N=Chance of nonoccurrence. Meaning that you must multiply the number of trials you will attempt by the theoretical chance of it happening, to the chance it will not happen, and find the square root of the product. If applying this to a die experiment you do 100 times, the formula would be as follows. SD= Square root of 100 x .20 x .80. This in the end will result in 4, meaning that most likely the experiment will be off somewhere in the range of four away from a perfect 20%. Note: The Central Limit Theorem states that there is a 68% chance of the experimental data being 1 SD away, 95% chance of 2 SDs away, and 99.7% chance of being 3SDs away.