Probability Review

	Belief in afterlife
Taken a college science class	Yes	No	Total
Yes	2702	634	3336
No	3722	837	4559
Total	6424	1471	7895

Data from General Social Survey

\(P(\text{belief in afterlife})\) = ? \(P(\text{belief in afterlife and taken a college science class})\) = ?
\(P(\text{belief in afterlife given taken a college science class})\) = ?

Calculate these probabilities and write them using correct notation. Use \(A\) for belief in afterlife and \(B\) for college science class.

Marginal Probability

	Belief in afterlife
Taken a college science class	Yes	No	Total
Yes	2702	634	3336
No	3722	837	4559
Total	6424	1471	7895

Data from General Social Survey

\(P(\text{belief in afterlife})\) = ?
\(P(A) = \frac{6424}{7895}\)

\(P(A)\) represents a marginal probability. So do \(P(B)\), \(P(A^C)\) and \(P(B^C)\). In order to calculate these probabilities we could only use the values in the margins of the contingency table, hence the name.

Joint Probability

	Belief in afterlife
Taken a college science class	Yes	No	Total
Yes	2702	634	3336
No	3722	837	4559
Total	6424	1471	7895

Data from General Social Survey

\(P(\text{belief in afterlife and taken a college science class})\) = ? \(P(A \text{ and } B) = P(A \cap B) = \frac{2702}{7895}\)

\(P(A \cap B)\) represents a joint probability. So do \(P(A^c \cap B)\), \(P(A\cap B^c)\) and \(P(B^c\cap B^c)\).

Note that \(P(A\cap B) = P(B\cap A)\). Order does not matter.

Conditional Probability

	Belief in afterlife
Taken a college science class	Yes	No	Total
Yes	2702	634	3336
No	3722	837	4559
Total	6424	1471	7895

Data from General Social Survey

\(P(\text{belief in afterlife given taken a college science class})\) = ? \(P(A \text{ given } B) = P(A | B) = \frac{2702}{3336}\)

\(P(A|B)\) represents a conditional probability. So do \(P(A^c|B)\), \(P(A | B^c)\) and \(P(A^c|B^c)\). In order to calculate these probabilities we would focus on the row or the column of the given information. In a way we are reducing our sample space to this given information only.

Note on conditional probability

\(P(\text{attending every class | getting an A}) \neq\) \(P(\text{getting an A | attending every class})\)

The order matters!

Complement of an Event

\(P(A^C)\) is called complement of event A and represents the probability of selecting someone that does not believe in afterlife.