# Probability Topics

## Independent and Mutually Exclusive Events

Independent and mutually exclusive do not mean the same thing.

### Independent Events

Two events are independent if the following are true:
1. P(A|B) = P(A)
2. P(B|A) = P(B)
3. P(A AND B) = P(A)P(B)
Two events A and B are independent events if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are not independent, then we say that they are dependent events.
Sampling may be done with replacement or without replacement.
1. With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
A bag contains four blue and three white marbles. James draws one marble from the bag at random, records the color, and replaces the marble. The probability of drawing blue is 
 
  . When James draws a marble from the bag a second time, the probability of drawing blue is still 
 
  . James replaced the marble after the first draw, so there are still four blue and three white marbles.
1. Without replacement: When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.
The bag still contains four blue and three white marbles. Maria draws one marble from the bag at random, records the color, and sets the marble aside. The probability of drawing blue on the first draw is 
 
  . Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now three blue and three white marbles. So, the probability of drawing blue is now 
 
  . Removing the first marble without replacing it influences the probabilities on the second draw.
If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise.

### Mutually Exclusive Events

A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0.
For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}. A AND B = {4, 5}. P(A AND B) =  and is not equal to zero. Therefore, A and B are not mutually exclusive. A and C do not have any numbers in common so P(A AND C) = 0. Therefore, A and C are mutually exclusive.
If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms.

### References

Gallup. (n.d.). Retrieved from www.gallup.com/
Lopez, S., and Sidhu, P. (2013, March 28). U.S. teachers love their lives, but struggle in the workplace. Gallup Wellbeing. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx

### Chapter Review

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent
In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.

### Formula Review

If A and B are independent, P(A AND B) = P(A)P(B), P(A|B) = P(A), and P(B|A) = P(B).
If A and B are mutually exclusive, P(A OR B) = P(A) + P(B) and P(A AND B) = 0.

### Homework

Use the following information to answer the next 12 exercises. The graph shown is based on more than 170,000 interviews that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Health Index Scores are the sample space. We randomly sample one type of Health Index Score, the emotional well-being score.

### Bringing It Together
