381 ProbabilityI Introduction to Probability QSCI 381 Lecture

381 Probability-I (Introduction to Probability) QSCI 381 – Lecture 7 (Larson and Farber, Sect 3. 1+3. 2)

381 Probability is to Statistics like a Bat is to Baseball!

Probability Experiments 381 n A is an action or trial, through which specific results (counts, measurements, or responses) are obtained. The result of a single trial in a probability experiment is an. The set of all possible outcomes of a probability experiment is the. An consists of one or more outcomes and is a subset of the sample space.

Probability 381 n Probability n n n Implies chance and uncertainty How do we measure it? How do probabilities behave?

Probability Experiments (Examples-I) 381 Probability Experiment Count of the number of trees in a stand Sample Space {0, 1, 2, …. } Event >10 Outcome 5

Probability Experiments (Examples-II) 381 Probability Experiment Sample Space Determine the maturity state of an baleen whale Mature, Calf, Juvenile Event Immature = {Juvenile OR Calf} Outcome Calf

Probability Experiments (Simple Events) 381 n n n An event that consists of only one outcome is called a (or an elementary event) Obtaining a maturity state of “mature” is a simple event. An event of a count of 5 or less trees in a stand is not a simple event because it consists of 6 possible outcomes: {0, 1, 2, 3, 4, 5}.

Classical Probability-I 381 n The (or theoretical) probability is used when each outcome in the sample space is equally likely to occur. The classical probability for an event E is given by: The probability of the event E.

Classical Probability-II (Example) 381 n n What is the (classical) probability of selecting the King of Clubs from a pack of 52 cards? If you drew a card from a pack and it was the King of Clubs, what is the probability of then randomly drawing another club?

Empirical Probability 381 n (or statistical) probability is based on observations obtained from probability experiments. The empirical probability of event E is the relative frequency of event E, i. e.

Empirical Probability (Examples) 381 n A lake consists of male and female fish. Males and females are equally susceptible to capture. You sample (and then release) 60 animals, of which 40 are female. n n What event are we interested in? What is the probability of the event? What is the probability of catching a female the next time you fish? How important are the assumptions about equal susceptibility and releasing fish?

The Law of Large Numbers 381 n As an experiment is repeated over and over, the empirical probability of an event approaches theoretical (actual) probability of the event

Subjective Probability 381 n n n probabilities arise from intuition, educated guesses and estimates. The probability that the Seahawks will win the Super Bowl next year is …. It is sometimes not that easy to distinguish between subjective and empirical probabilities.

Probabilities Formalized 381 n n n A probability cannot be negative and cannot exceed 1, i. e. : The probability of an event in the sample space (the set of all possible outcomes) is 1. The complement of the event E is the set of outcomes not part of E. The probability of the complement of E (denoted E’ , E-prime) is:

Examples 381 n 40 fish are sampled from a lake with a large population. If 29 are female, what is the probability of sampling a male? n n Probability of sampling a female, P(E) = ? Probability of sampling a male is 1 -P(E) = ? If the lake only had 80 fish and the sex ratio was 50: 50 initially, what is the probability of the 41 st fish being a male? Genetics: Two snapdragons (Red and White) are crossed, the possible outcomes are RR, RW, WR and WW. What is the probability of the events: Red, Pink (=white+red), and White?

Example (from a Test) 381 The sex ratio at birth for a particular pinniped is 55: 45 male: female. You select two pups at random from the population. What is the probability that: n n n a. they are both male? b. at least one of them is male? Solution: n n We can assume that the population is “large” (not 100!) P(male) * P(male) = 0. 55*0. 55 P(at least one male) = 1 -P(both female) = 1 - 0. 45*0. 45

Conditional Probability-I 381 n A is the probability of an event occurring given that another event has already occurred. The conditional probability of event B occurring given that event A has already occurred is denoted P(B|A) and is read as “probability of B given A”

381 Conditional Probability-II Gene Present Gene Not Present Total High IQ 33 19 52 Normal IQ 39 11 50 Total 72 30 102 • What is the probability of having a high IQ? • What is the probability of having a high IQ if you have the gene?

381 Conditional Probability-III Gene Present Gene Not Present Total High IQ 33 19 52 Normal IQ 39 11 50 Total 72 30 102 • What is the probability of having a high IQ? • Solution: This question says nothing about the other factor (gene) so we look at the last column (total); =52/102

381 Conditional Probability-IV Gene Present Gene Not Present Total High IQ 33 19 52 Normal IQ 39 11 50 Total 72 30 102 • What is the probability of having a high IQ if you have the gene? • Solution: The question relates only to having the gene so we are dealing with a conditional probability. We focus on the “Gene present” column and find the probability of having a High IQ GIVEN having the gene = 33/72.