# Math 225

## Introduction to Biostatistics

### Chapter 3

1. Probability is the branch of mathematics quantifies uncertainty.

2. Probabilites are measured on a scale from zero to one where events with no chance of occuring have probability zero and events that are certain to occur have probability one.

3. Probability may be thought of as the long-run relative frequency with which a particular event would occur if the same random experiment could be repeated indefinitely. This viewpoint is the frequentist interpretation of probability.

4. Another school of thought views probability as a measure of the strength of belief. This explicitly subjective viewpoint of probability is called Bayesian.

5. The set of all possible outcomes in a random experiment is called the outcome space.

6. An event is a subset of the outcome space.

7. If we assume that all of the outcomes are equally likely, then the probability of an event is the number of outcomes in the event divided by the number of outcomes in the outcome space.

8. For example, in rolling a single die, the outcome space is the set {1,2,3,4,5,6}.

The event "a 1 is rolled" has probability 1/6 while the event "an even number is rolled" has probability 3/6.

9. Two events are mutually exclusive if they have no outcomes in common. In other words, if one event occurs, the other cannot possibly occur. The events "a 1 is rolled" and "an even number are rolled" are a pair of mutually exclusive events.

10. Addition Principle: If A and B are mutually exclusive events, then P(A or B) = 0.

Here, "A or B" is the event that either A occurs or B occurs or both occur. It does not mean that exactly one of A or B occur.

11. Two events are independent if the outcome of one does not affect the probability of the outcome of the other.

12. Multiplication Principle: If A and B are independent events, then P(A and B) = P(A) * P(B).

Here, "A and B" is the event of outcomes in both A and B. The symbol "*" means multiplication.

13. Inclusion-Exclusion: If A and B are not mutually exclusive, there is another formula for the probability of their union. This formula is always correct.

P(A or B) = P(A) + P(B) - P(A and B)

In calculating P(A), we count all outcomes in A. In calculating P(B), we count all outcomes in B. We have overcounted all outcomes in both A and B, so get the correct probability by subtracting these off.

14. Counting (enumeration) problems often involve these types of calculations.

1. Factorials. The number of ways to arrange k labeled items in is k! = k*(k-1)*(k-2)* ... * 1 if k is a positive integer.

Because there is exactly one way to arrange zero items in order, 0!=1.

2. Permutations. The number of ways to select r items from k labeled items in order is kPr = k*(k-1)*(k-2)* ... *(k-r+1) if k and r are positive integers and r is not greater than k.

There are r factors in this product.

Notice that kPk = k!.

Also, in general, kPr = k! / (k-r)!.

3. Combinations:. The number of ways to choose r items from k labeled items without regard to order is kCr = k! / (r! * (k-r)!).

Notice that kCr = kPr / r!. There are r! ways to place the r chosen items in order after they have been selected.

15. Example. If a box has ten balls, five black and five white and three balls are drawn out at random without replacement, what is the probability of drawing out three black balls?

Solution 1: There are 10P3 = 10*9*8 = 720 ways to choose three of the ten balls in order. There are 5P3 = 5*4*3 = 60 ways to choose three of the five black balls in order. Thus, the probability is 60/720 = 1/12.

Solution 2: There are 10C3 = 10*9*8/(3*2*1) = 120 ways to choose three of the ten balls without regard to order. There are 5C3 = 5*4*3/(3*2*1) = 10 ways to choose three of the five black balls without regard to order. Thus, the probability is 10/120 = 1/12.