Counting And Probability

MULTIPLICATION PRINCIPLE, FACTORIAL, PERMUTATION

Basic Probability

Probability is the chance that a certain event will happen. Usually probability is expressed in a ratio.

How about starting with a simple example?

When a coin is tossed, there are two possibilities either heads or tails. The probability of the coin landing on heads is 1/2.


When a dice is thrown, there are 6 possibilities: 1,2,3,4,5,6

The probability of any one of them is 1/6.


Therefore, the probability of an event happening is: (the number of favorable cases)/(the number of possible cases)

Example: There are five marbles in a box: 4 are blue and 1 is red..

What is the probability of getting a blue marble?

4/5 = 0.8

Note: The probability is always between 0 and 1.

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Let the number of possible outcomes of an event be n(E).

If the two possible events E1 and E2 have no common outcomes then n(E)=n(E1)+n(E2)

If event E1 can result in any one of n(E1) possible outcomes, and for each outcome of the event E1, there are n(E2) possible outcomes of E2 then n(E)=n(E1) x n(E2)

Permutation

A permutation is an arrangement of a group of objects in a specific order. Consider 4 numbers 1,2,3,4 to be arranged in different orders. Since each number can only be used once, then the first space can be occupied by 4 different numbers, the second space by 3, the third space by two and the last space by 1.

So the total number of permutations that can be made is: 4x3x2x1=4!=24

The total number of permutations that can be formed from n objects without repetition is n! which is read n factorial.

The symbol nPr is the number of permutations formed from n objects taken r at a time.

Consider 6 different books taken 3 at a time... 6P3=120

In other words, this is expressed as (6!)/(6-3)! = 120


But this is not the only case.. What about permutations of objects that are NOT different?

The four numbers 1,6,6,4 can be arranged in 4!=24 ways. But because two numbers are the same then some of the possibilities will be alike.

4! = 24

2! = 2

Therefore the number of permutations is 12.

How many different permutations can be made using all the letters of the word Connecticut?

-- It contains 11 letters including 3 C's, 2 N's, and 2 T's.

The number of permutations is:

11! = 39916800

3! 2! 2! = 24

Therefore, the number of permutations is 1,633,200

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