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Section 5.1-5.2

Section 5.1-5.2. Probability Addition Rule and Complements . Subjective Probability. A subjective probability of an outcome is a probability obtained on the basis of personal judgment. Probability of an outcome . Is the likelihood of observing the outcome

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Section 5.1-5.2

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  1. Section 5.1-5.2 Probability Addition Rule and Complements

  2. Subjective Probability • A subjective probability of an outcome is a probability obtained on the basis of personal judgment.

  3. Probability of an outcome • Is the likelihood of observing the outcome • High likelihood of happening is probability close to 1 • Low likelihood of happening is probability close to 0

  4. Probability • Can be expressed as decimals, fractions, or percents. • To change a decimal to a fraction, read it, write it as a fraction and reduce it. • To change a fraction to a decimal, divide the numerator by the denominator. • To change a decimal to a percent, move the decimal point 2 places to the right and add %

  5. Law of Large Numbers • As the numbers of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.

  6. Experiment • Any process with uncertain results that can be repeated.

  7. Sample Space • Known as S • The sample space of a probability experiment is the collection of all possible outcomes.

  8. Event • An event is any collection of outcomes from a probability experiment. • May consist of one outcome, called simple events, e sub I, or more than one outcome. • Events are denoted using capital letters such as E.

  9. On a roll of a fair die: • The outcomes of the probability of the experiment are : e sub1 ={1}, e sub 2 ={2}, e sub 3 = {3}, e sub 4 = {4} , e sub 5={5}, e sub 6 = {6} because these are the outcomes when you roll a die.

  10. On the Roll of a fair die: • The sample space will be written: S= {1,2,3,4,5,6}

  11. The event E={roll of an odd number} • E={1,3,5}

  12. Rules of Probabilities • 1. The probability of an event occurring, E, or P(E), must be greater than or equal to 0 and less than or equal to 1. • 0 ≤ P(E) ≤1

  13. Rules of Probabilities • 2. The sum of the probabilities of all outcomes must equal 1. That is if the sample space, S ={esub1 + e sub 2 +e sub3 +… e sub n} then P(e sub 1) +P(e sub 2) +… +P(e sub n )=1.

  14. A probability model… • Lists the possible outcomes of a probability experiment and each outcomes probability. • Must satisfy rules 1 and 2 of the rules of probability.

  15. Probability • If the probability of an event = 0, Then the event is impossible • If the probability of an event =1 then the event is a certainty.

  16. Unusual Event • An unusual event is an event that has a low proportionality of occurring. • An unusual event will have a probability of less than 0.05 or 5% usually, but this is not necessarily the cutoff point. • The cutoff point is determined by the situation. • Statisticians use .01, .05, and .1 typically.

  17. 3 methods for determining the P(e) • 1. Empirical Method • 2. Classical Method • 3. Subjective Method

  18. Empirical Method • The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment. • P(E)≈frequency of E/ number of trials of experiment • Note: this is an approximation

  19. Classical Method • If an experiment has n likely outcomes and if the number of ways that an event can occur is m, then the probability of E, P(E), is • P(E) = m/n • or number of successes/total # of outcomes.

  20. A Pair of dice are rolled… • Compute the probability of rolling a number 7. • There are 6 x 6 or 36 possible outcomes for the rolling the dice. • Set up a tree diagram of the possible outcomes. • Find that there are 6 possible ways to roll a 7. • P(E)=6/36=1/6=.16 with the repetend bar over 6.

  21. Probability of rolling a 3: • Again there are 36 outcomes and rolling a 3 can happen twice. So the probability will be • 3/36 = 1/12= .83 with 3 repeating

  22. Disjoint • 2 events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. • If E and F are disjoint events, then P(E or F)= P(E) +P(F)

  23. Complement of an Event • Let S be the sample space of a probability experiment and let E be the event. The Complement of E, denoted as E superscript c, is all outcomes that are not outcomes o

  24. The General Addition Rule • For any two events E and F, • P(E or F) = P(E)+P(F) – P(E and F)

  25. Complement Rule • If E represents any event and E superscript c represents the complement of E, then • P(E superscript c)= 1 - P(E)

  26. Homework: • Problems 1-10, and 11-41 odd pages 269-271 • Problems 1-4, and 5-43 odd pages 281-285

  27. Have a great evening. Questions?

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