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Stat 528 (Autumn 2008) Probability
Reading: Sections 4.1, 4.2, 4.5.
- Chance experiments
- Venn diagrams and events
- Complements, and, or
- Empty and disjoint events
- Probabilities
- The complement rule
- Addition rule for disjoint events
- The general addition rule
- Independence (the multiplication rule)
- Conditional probability
- The law of total probability
- Bayes Theorem
Chance experiments and events
- A chance experiment or a random experiment is an activity in which the result may change if the experiment is repeated many times – it depends on chance. e.g., tossing a coin, rolling a die, drawing a card from a well- shuffled deck, seeing whether it rains tomorrow, etc.
- It arises because:
- some natural phenomenon is at work;
- we introduce randomness (e.g., in an experimental or survey design).
- An outcome is a possible result of a chance experiment.
- The sample space, S, is the set of all possible outcomes in the experiment.
- An event is an outcome or collection of outcomes of a chance experiment. An event is a subset of the sample space. We use letters A, B, C,... to denote events.
Complement
- For an event A the complement of A, Ac^ consists of all events in S, that are not in A. Sample space, S
A Ac
Intersection
- The intersection of two events A and B, A and B, con- sists of all events common to both A and B. Sample space, S A A and B B
- A and B are disjoint or mutually exclusive if they have no events in common. A and B is the empty set, ∅. Sample space, S
A B
An example
Suppose we take a random sample of six batteries from a produc- tion line. Let
A = {there are more than three defective batteries in the random sample}; B = {there are fewer than five defective batteries in the random sample}.
Assigning probabilities
- We assign probabilities to events. We let P (A) denote the probability of the event A.
- The two main interpretations of probability are :
- long-run relative frequency
- subjective assessment
- The rules (axioms) of probability: Suppose we have a chance experiment with sample space S. Then (1) 0 ≤ P (A) ≤ 1 for any event A. (2) P (S) = 1. (3) If A and B are disjoint events, P (A or B) = P (A) + P (B).
Standard situations
- The flip of a fair coin. The sample space is S = {H, T }, and P (H) = P (T ) = ____.
- The roll of a six sided die. The sample space is S = { 1 , 2 , 3 , 4 , 5 , 6 }. If the die is fair we have = P (roll a 1) = P (roll a 2) = P (roll a 3) = P (roll a 4) = P (roll a 5) = P (roll a 6) = ____.
Thus P (get any number on the die) = P (S) = 1, and P (get an even number on the die) = 3/6.
The complement rule
- For any event A, P (Ac) = 1 − P (A)
- Ex: For the six sided die example, consider the event A = {the die roll is an even number}. What is the probability of Ac?
More dice games
For the die example define three events:
A = {roll a 2, 4 or 6}, B = {roll a 1, 2 or 3}, and C = {roll a 5}.
Independence
- Events A and B are independent events if the probabil- ity of either one occurring is not affected by the other event occurring: P (A and B) = P (A)P (B).
- This is also called the multiplication rule for indepen- dent events.
- Implications: if A is independent of B then
- Ac^ is independent of Bc,
- A is independent of Bc, and
- Ac^ is independent of B.
Using independence
- The model for a pair of fair dice, one red, one green.
More on independence
- Independence comes in two main styles.
- Structural independence: Events are independent because the mathematical model used to create the probabilities forces independence. - Roll two fair dice, one red, one green.
A = {roll a 2, 4 or 6 on the red die}, B = {roll a 1, 2 or 3 on the green die}.
- Accidental independence: Events satisfy the formal definition of independence, but a small change to the model destroys the independence. - Roll two fair dice, one red, one green.
C = {roll a 2 on the red die}, D = {roll a total of 7 on the two dice}.
Conditional probability
- The rules for working with dependent events are much like those for working with independent events. Complements and unions work in exactly the same fashion. Intersections work in a similar fashion. The key idea is conditional proba- bility.
- The conditional probability of event B given event A is P (B|A) P (B|A) = P^ (A P^ and(A)^ B), provided P (A) 6 = 0.
- The conditional probability of B given A is the probability of the event B occurring, given the knowledge that A has occurred.
The multiplication rule
- The multiplication rule follows from the definition of condi- tional probability: P (A and B) = P (A)P (B|A).
- The law of total probability can be used to compute the prob- ability of an event. P (A) = P (A and B) + P (A and BC^ ) = P (B)P (A|B) + P (BC)P (A|BC).
