Probability Rules

Now, we are going to provide a mathematical treatment of probability. That means we are going to introduce formal definitions, axioms, and theorems. We illustrate these with examples from one coin toss and the roll of one die.

Expression | Definition | One Coin Toss | Sum of Two Dice |
---|---|---|---|

S | The Sample Space | {H,T} | {2,3,4,5,6,7,8,9,10,11,12} |

P(A) | The Probability of Event A | P(H) = 1/2 | P(4) = 3/36 |

A^{c} |
The Complement of Event A | H^{c} = T |
4^{c} = {2,3,5,6,7,8,9,10,11,12} |

Next, we introduce a few simple axioms.

Axiom | Explanation | Coin Example | Dice Example |
---|---|---|---|

P(S) = 1 | One of the outcomes in the sample space must occur. | must be H or T | must be 2 - 12 |

P(A) >= 0 | Probability is never less than 0 | P(H) > 0 | P(4) > 0 |

From these definitions and axioms, it follows that P(A^{c}) = 1 - P(A). That is, if A does not occur, then the complement of A must occur. If a coin flip is not heads, then it is tails. If a dice roll is not 4, then it must be 2,3, or 5-12.

Compound Events

A compound event is an event that is derived from two other events. For example, if we roll two dice, then the event "getting a six on either the first or second die" is a compound event.

We could flip a coin and roll a die to get a compound event. The compound event might be (H,4), meaning that the coin came up heads and we rolled a 4 on the die.

There are two types of compound events:

- The union of two events A and B is the probability of A or B occuring. It is written as P(A or B).
- The intersection of two events A and B is the probability of A and B occuring. It is written as P(A and B).

Two events are said to be disjoint, or mutually exclusive, if and only if P(A and B) = 0. For example, if we roll one die and event A is getting a 2 and event B is getting a 3, then it is impossible for the event "A and B" to occur. The two events are mutually exclusive.

Two events are said to be independent if P(A and B) = P(A)P(B), provided that P(A) and P(B) are both nonzero. What that means is that the probability of one event occuring does not depend on whether or not the other event occurs. If A is getting heads on a coin flip and B is rolling a 4 on a die, then A and B are independent.

Consider a single roll of a die. Define event A as getting a number greater than 3. Define event B as getting an even number. Are A and B mutually exclusive? Are A and B independent?

QUESTIONS:

- Give an example of a pair of events A and B that are disjoint but not independent.
- Give an example of a pair of events A and B that are independent but not disjoint.
- Is it possible for events to be both independent and disjoint? Why or why not?
- If A and B are independent, do you think that A
^{c}and B are necessarily independent?

There is a general rule for finding the union of two events.

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

A fairly typical exam question is to give you three out of the four probabilities in this equation and ask you to solve for the fourth. Note that if A and B are disjoint, then the last term is zero, and the union of the probabilities is just P(A) + P(B).

For the compound event A and B, we can think of the sample space as consisting of four elements:

{(A and B), (A^{c} and B), (A and B^{c}), (A^{c} and B^{c})}

A fairly typical exam question is to give you the probabilities for three out of the four of these elements and then ask you to solve for the fourth. Once you remember that the probabilities for all four have to sum to one--because this is the sample space--it becomes pretty simple.