Chapter 4 Probability

Random phenomenon A phenomenon is random if individual outcomes are uncertain, but has a regular distribution of outcomes in a large number of repetitions. Probability is the likelihood of an individual outcome. Examples: Tossing a coin, rolling a die, choosing a card

4.2 Probability Models The sample space S of a random phenomenon is the set of all possible outcomes. Find the sample space for Rolling a die Tossing two coins Rolling two dice Probability Models list the sample space and probabilities.

Probability Models Properties of Probability Models Probabilities are between 0 and 1 The sum of probabilities of all possible outcomes is 1 Probability an even does not occur is 1 – probability it does occur P(AC)=1-P(A) Event – an event is an outcome or set of outcomes of a random phenomenon. Disjoint events – events with no outcomes in common

Probability Models Example (choosing one card): A –King B – Queen C – Hearts If two events are disjoint, then, P(A or B)=P(A) + P(B) (referred to as the addition rule)

Studying a language other than English (grades 9-12) Language English French Asian/Pacific Other Probability 0.59 0.23 0.07 ? What probability should replace “?” in the distribution? What is the probability that a Canadian’s mother tongue is not English?

Probability Equally likely outcomes: If a random phenomenon has K possible outcomes, all equally likely, then each individual outcome has prob 1/K. The probability of any event A is P(A) = (# outcomes in A)/(total # outcomes in S)

Examples Rolling two dice Family having 3 children Probability of rolling at least one 6. Probability of rolling a sum less than 4. Family having 3 children Probability of having all boys Probability of exactly one boy Probability of having at least one boy

Probability Independent events: Two events A and B are independent if knowing that one occurs does NOT change the probability that the other occurs. Examples Tosses of a coin Individuals in a random sample If two events are independent, then P(A and B) = P(A)P(B) (Multiplication Rule)

Probability Example: Tossing a coin 8 times, what is the probability of getting all 8 heads? Try the next problem (info from the 1990 census) A is the event that a randomly chosen American is Hispanic B is the event that a randomly chosen American is white

Table for problem #4.20 A Hispanic Not Hispanic Total Asian 0.000 0.036 Black 0.003 0.121 0.124 White 0.06 0.691 0.751 Other 0.062 0.027 0.089 0.125 0.875 1 B

Problem 4.22 Verify that the table gives a legitimate assignment of probabilities. What is P(A)? Describe BC in words and find P(BC) by the complement rule. Express “the person chosen is non-Hispanic white” in terms of events A and B. What is the probability of this event?

4.3 Random Variables A random variable is a variable whose value is a numerical outcome of a random phenomenon. Example: A family with 3 children S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG} Define a random variable Y = # of girls, then the probability model for Y is:

Random Variables Probability Model for Y (assuming equally likely outcomes): y P(y) 0 1/8 1 3/8 2 3/8 3 1/8

Random Variables Another example: Tossing two dice Define Y = sum of two dice, find probability model for Y. A discrete random variable has A finite # of possible values. Every probability must be between 0 and 1 Sum of probabilities of all outcomes is 1

Random variables Histograms displaying probabilities (relative frequencies) are referred to as probability histograms

Random Variables Now, let’s take a look at #4.55

Random Variables Continous random variables can take on infinitely many values within an interval (example: height, weight, length of rope) Probability distributions for continuous random variables are described by density curves. Probability is defined as the area under the curve.

Continuous Random Variables Illustrate some density curves. Now, Probability of event is the area under the curve. For example, if P(A)=P(a<X<b) We will take a look at two continuous distributions: Uniform distribution Normal distribution