Chapter 4 Probability, Randomness, and Uncertainty

Section 4.1 Classical Probability

Vocabulary Probability Experiment (trial) – any process in which the result is random in nature Outcome – each individual result that is possible Sample Space – the set of all possible outcomes for a given probability experiment Event – a subset of outcomes of the sample space

Tree Diagram

Types of Probability Subjective An educated guess regarding the chance that an event will happen Empirical Experimental, probability of whatever happens in the experiment Classical Theoretical, probability of what is supposed to happen

Empirical

Classical Law of Large Numbers – the larger the number of trials, the closer the empirical probability will be to the true probability (classical)

NOTE: Watch for problems with the words “at least” and “at most”

Section 4.2a Probability Rules: Properties, the Complement, and Addition Rules

Facts about Probability 1. Each outcome in the event has a probability between 0 and 1, inclusive. 2. The probability that an event is certain is The probability that an event will not happen is 0

The Complement The probability of the complement is all outcomes in the sample space that is not in E. Example: What is the complement of rolling an odd number? Solution: An even number. (2, 4, 6)

Complement

Addition Rule The probability of event E or event F happening is the probability of event E plus the probability of event F minus the probability of event E and event F. The probability of event E and event F is what the two events have in common.

Mutually Exclusive Events that cannot happen at the same time. They do not have anything in common.

Section 4.2b Independence, Multiplication Rules, and Conditional Probability

Multiplication Rules “and” problems With repetition With replacement Without repetition without replacement Independent – one event has no effect on the other Dependent – one event does have an effect on the other

Independent Events If two events are independent, we multiply the probabilities of each event.

Conditional Probability

Dependent Events The probability that event E and event F happening is equal to the probability of event E multiplied by the probability of event F happening given that event E happened first. The second probability is the conditional.

Section 4.3 Counting Rules

Fundamental Counting Principle States that you can multiply together the number of possible outcomes for each stage in an experiment in order to obtain the total number of outcomes for that experiment.

Special Combinations and Permutations

Section 4.4 Additional Counting Techniques