Chapter 5

Factorial Symbol

If n> or equal to 0 is an integer, the factorial function (symbol: !) means to multiply a series of descending natural numbers. Examples: 4! = 4 × 3 × 2 × 1 = 24.

General Multiplication Rule

P(E and F) = P(E) x P(F|E)

Empirical Rule

P(E)= frequency of E/number of trials

Conditional Probability Rule

P(F|E) = P(E and F)/P(E) = N(E and F)/N(E) The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the number of outcomes in E. *"given that"

Define at least

greater than or equal to

Combination

is a collection, without regard to order, in which r objects are chosen from n distinct objects with r< or equal to n without repetition. The symbol nCr represents the number of combinations of n distinct objects taken r at a time.

Combination formula

nCr = n!/r!(n-r)!

Independent event

that one event occurring does not affect the probability of the other event occurring. *with replacement

Conditional Probability

the notation P(F|E) is read "the probability of event F given event E." It is conditional probability that event F occurs, given that event E has occurred.

Rule 2 of probability

the sum of the probabilities of all outcomes in the sample space must equal 1.

Rule 3 of Probability

If E and F are disjoint events, then P(E or F) = P(E) + P(F). If E and F are not disjoint events, then P(E or F) = P(E) -P(E and F).

Rule 5 of Probability

If E and F are independent events, then P(E and F) = P(E) x P(F)

What is the difference between the Addition Rule and Multiplication Rule?

Notice that or probabilities use the Addition Rule, whereas and probabilities use the Multiplication Rule. Accordingly, or probabilities imply addition, whereas and probabilities imply multiplication.

Which type of compound event is generally associated with multiplication? Which is generally associated with addition?

The word "AND" is generally associated with multiplication and "OR" is generally associated with addition.

Expressing Independence using conditional probabilities

Two events E and F are independent if P(E|F)=P(E) or equivalently, if P(F|E) = P(F).

Unusual Result?

events with probability less than 0.05 (5%) are unusual.

Dependent Event

Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F.

Disjoint Event (Mutually Exclusive)

Two events are disjoint if they have no outcomes in common, that is, knowing that one of the events occurs, we know that the other event did not occur. *Therefore knowing two events are disjoint means that the events are not independent.

Multiplication rule for Independent Events

If E and F are independent events, then P(E and F) = P(E) x P(F) Can also be extended to 3 or more independent events.

Complement Rule *Usually, when computing probabilities involving the phrase at least...

If E represents any event and E^C represents the complement of E, then P(E^C)= 1-P(E)

Independence and small samples

If small random samples are taken from large populations without replacement, it is reasonable to assume independence of the events. As a rule of thumb, if the same size, n, is less than 5% of the population size, N, we treat the events are independent. That is, if n < 0.05N, treat the events as independent.

Permutation

is an ordered arrangement of r objects chosen from n distinct objects without repetition.

What method of assigning probabilities to a simple event uses relative frequencies?

The empirical method - approximates the probability of an event E by dividing the number of times event E is observed by the number of repetitions of the experiment. This is the relative frequency of event E. P(E) = relative frequency of E = frequency of E/number of trials of experiment.

Rule 1 of probability

The probability of any event must be between 0 and 1, inclusive. If we let E denote any event, then 0 is < or equal to P(E) is < or equal to 1.

If n> or equal to 0 is an integer, the factorial symbol n!, is defined by the formulas...?

n!=n(n-1) x3x2x1 1!=1 0!=1

Number of Permutations Formula

nPr=n!/(n-r)! 1. The n objects are distinct 2. Repetition of objects is not allowed, and 3. Order is important (so ABC is different from BCA)