Probability Refresher

You may want to read the Wikipedia article on Probability or some Khan Academy video mini-lectures on probability.

#### Basics to Remember

**Probability**is the likelihood that an**outcome**will happen (e.g., we will roll a 2 on a die).- The value of a probability can range from 0 (the event will certainly not happen) to 1 (the even will certainly happen).
- An "
**exhaustive**" listing of outcomes refers includes everything that could happen. - An
**event**is a set of outcomes (e.g., we roll an odd number - outcomes roll a 1, roll a 3, roll a 5). - Two events are "
**mutually exclusive**" if the occurrence of one precludes the other. Another way to say this is that they have no outcomes in common.- Rolling an odd number and rolling an even number are mutually exclusive events. Rolling an odd number and rolling a number less than 3 are not.

- If the probability of an event A is P(A) then the probability of A not happening is $1 - P(A)$
- We say that two events are
**independent**if the occurrence of one does not affect the probability of the other. - If two events, A and B, are independent, then the probability that A AND B happen is
- $P (A \hspace{2 mm}and \hspace{2 mm}B) = P(A) \times P(B)$

- IF two events, A and B, are mutually exclusive, then the probability that A OR B happens is
- $P (A \hspace{2 mm}or \hspace{2 mm}B) = P(A) + P(B)$

- IF two events, A and B, are NOT mutually exclusive, then the probability that A OR B happens is
- $P(A) + P(B) - P (A \hspace{2 mm}and \hspace{2 mm}B)$

- The
**conditional probability of A on B**is the probability of event A,the occurrence of event B. This is one of the most important concepts in the social and policy sciences. Conditional probability is written $\mathrm{P}(A \mid B)$, and is read "the probability of A, given B." The value is given by**given**- $P(A|B) = \frac {P (A \hspace{2 mm}and \hspace{2 mm}B)} {P(B)}$

#### Review Questions Probability

You will need to be able to do these to complete this course successfully.

- What is the probability of throwing a 1 in a single toss of a single die?
- What is the probability of throwing a 2 in a single toss of a single die?
- What is the probability of the event of throwing something less than 3 (that is, 1
**OR**2) in a single toss of a single die? - What is the probability of the event of throwing two 1s in toss of two dies?
- What is the probability of the event of throwing something less than 4 (that is, 1
**OR**2**or**3) in a single toss of a single die? - What is the probability that you threw a 1 given that you threw an odd number in a single roll of a single die?
- What is the probability that you roll a total of 7 given that your first roll was a 3 when rolling two dice in sequence?
- What is the probability of the event of throwing something less than 4 (1,2,3) or something odd (1,3,5) in a single toss of a single die?