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Probability-I (Introduction to Probability)

Probability-I (Introduction to Probability). QSCI 381 – Lecture 7 (Larson and Farber, Sect 3.1+3.2). Probability is to Statistics like a Bat is to Baseball!. Probability Experiments. Probability experiment.

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Probability-I (Introduction to Probability)

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  1. Probability-I(Introduction to Probability) QSCI 381 – Lecture 7 (Larson and Farber, Sect 3.1+3.2)

  2. Probability is to Statistics like a Bat is to Baseball!

  3. Probability Experiments Probability experiment • 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 allpossible outcomes of a probability experiment is the . An consists of one or more outcomes and is a subset of the sample space. Outcome Sample space Event

  4. Probability • Probability • Implies chance and uncertainty • How do we measure it? • How do probabilities behave?

  5. Probability Experiments(Examples-I)

  6. Probability Experiments(Examples-II)

  7. Probability Experiments(Simple Events) • 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}. Simple event

  8. Classical Probability-I • 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: Classical The probability of the event E.

  9. Classical Probability-II(Example) • 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?

  10. Empirical Probability • (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

  11. Empirical Probability(Examples) • 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. • 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?

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

  13. Subjective Probability Subjective • 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.

  14. Probabilities Formalized • 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:

  15. Examples • 40 fish are sampled from a lake with a large population. If 29 are female, what is the probability of sampling a male? • 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 41st 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?

  16. Example (from a Test) • 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: • a. they are both male? • b. at least one of them is male? • Solution: • 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

  17. Conditional Probability-I • 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” Conditional probability

  18. Conditional Probability-II • What is the probability of having a high IQ? • What is the probability of having a high IQ if you have the gene?

  19. Conditional Probability-III • 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

  20. Conditional Probability-IV • 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.

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