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5.1

Probability Rules!. 5.1. Probability relates short-term results to long-term results An example A short term result – what is the chance of getting a proportion of 2/3 heads when flipping a coin 3 times A long term result – what is the long-term proportion of heads after a great many flips

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5.1

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  1. Probability Rules! 5.1

  2. Probability relates short-term results to long-term results • An example • A short term result – what is the chance of getting a proportion of 2/3 heads when flipping a coin 3 times • A long term result – what is the long-term proportion of heads after a great many flips • A “fair” coin would yield heads 1/2 of the time – we would like to use this theory in modeling Probability

  3. Relation between long-term and theory • The long term proportion of heads after a great many flips is 1/2 • This is called the Law of Large Numbers • Relation between short-term and theory • We can compute probabilities such as the chance of getting a proportion of 2/3 heads when flipping a coin 3 times by using the theory • This is the probability that we will study Long Term Probability

  4. Some definitions • An experiment is a repeatable process where the results are uncertain • An outcome is one specific possible result • The set of all possible outcomes is the samplespace • Example • Experiment … roll a fair 6 sided die • One of the outcomes … roll a “4” • The sample space … roll a “1” or “2” or “3” or “4” or “5” or “6” Definitions

  5. More definitions • An event is a collection of possible outcomes … we will use capital letters such as E for events • Outcomes are also sometimes called simpleevents … we will use lower case letters such as e for outcomes / simple events • Example (continued) • One of the events … E = {roll an even number} • E consists of the outcomes e2 = “roll a 2”, e4 = “roll a 4”, and e6 = “roll a 6” … we’ll write that as {2, 4, 6} Definitions Continued

  6. Summary of the example • The experiment is rolling a die • There are 6 possible outcomes, e1 = “rolling a 1” which we’ll write as just {1}, e2 = “rolling a 2” or {2}, … • The sample space is the collection of those 6 outcomes {1, 2, 3, 4, 5, 6} • One event is E = “rolling an even number” is {2, 4, 6} Die Rolling

  7. Rule – the probability of any event must be greater than or equal to 0 and less than or equal to 1 • It does not make sense to say that there is a –30% chance of rain • It does not make sense to say that there is a 140% chance of rain • Note – probabilities can be written as decimals (0, 0.3, 1.0), or as percents (0%, 30%, 100%), or as fractions (3/10) Probability…between 0 and 1

  8. Rule – the sum of the probabilities of all the outcomes must equal 1 • If we examine all possible cases, one of them must happen • It does not make sense to say that there are two possibilities, one occurring with probability 20% and the other with probability 50% (where did the other 30% go?) Sum of Probabilities

  9. Probability models must satisfy both of these rules • There are some special types of events • If an event is impossible, then its probability must be equal to 0 (i.e. it can never happen) • If an event is a certainty, then its probability must be equal to 1 (i.e. it always happens) Special Events

  10. A more sophisticated concept • An unusualevent is one that has a low probability of occurring • This is not precise … how low is “low? • Typically, probabilities of 5% or less are considered low … events with probabilities of 5% or lower are considered unusual That’s Unusual?!

  11. Probability • If we do not know the probability of a certain event E, we can conduct a series of experiments to approximate it by • This becomes a good approximation for P(E) if we have a large number of trials (the law of large numbers)

  12. Example • We wish to determine what proportion of students at a certain school have type A blood • We perform an experiment (a simple random sample!) with 100 students • If 29 of those students have type A blood, then we would estimate that the proportion of students at this school with type A blood is 29% Probability

  13. We wish to determine what proportion of students at a certain school have type AB blood • We perform an experiment (a simple random sample!) with 100 students • If 3 of those students have type AB blood, then we would estimate that the proportion of students at this school with type AB blood is 3% • This would be an unusual event Example (Continued)

  14. The classical method applies to situations where all possible outcomes have the same probability • This is also called equallylikelyoutcomes • Examples • Flipping a fair coin … two outcomes (heads and tails) … both equally likely • Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … all equally likely • Choosing one student out of 250 in a simple random sample … 250 outcomes … all equally likely Equally Likely Outcomes

  15. Because all the outcomes are equally likely, then each outcome occurs with probability 1/n where n is the number of outcomes • Examples • Flipping a fair coin … two outcomes (heads and tails) … each occurs with probability 1/2 • Rolling a fair die … six outcomes (1, 2, 3, 4, 5, and 6) … each occurs with probability 1/6 • Choosing one student out of 250 in a simple random sample … 250 outcomes … each occurs with probability 1/250 Equally Likely Outcomes

  16. The general formula is • If we have an experiment where • There are n equally likely outcomes (i.e. N(S) = n) • The event E consists of m of them (i.e. N(E) = m) Classical Probability

  17. Because we need to compute the “m” or the N(E), classical methods are essentially methods of counting • These methods can be very complex! • An easy example first • For a die, the probability of rolling an even number • N(S) = 6 (6 total outcomes in the sample space) • N(E) = 3 (3 outcomes for the event) • P(E) = 3/6 or 1/2 Classical Probability

  18. A more complex example • Three students (Katherine, Michael, and Dana) want to go to a concert but there are only two tickets available • Two of the three students are selected at random • What is the sample space of who goes? • What is the probability that Katherine goes? Example

  19. Katherine Start Michael Dana First ticket • Example continued • We can draw a treediagram to solve this problem • Who gets the first ticket? Any one of the three… Tree Diagram

  20. Katherine Second ticket Start Michael Michael Firstticket Dana Dana • Who gets the second ticket? • If Katherine got the first, then either Michael or Dana could get the second Tree Diagram

  21. Outcomes Katherine Michael Katherine Dana Katherine Secondticket Start Michael Michael Dana Dana Firstticket That leads to two possible outcomes Example

  22. Katherine Start Michael Katherine Katherine Michael Michael Dana Dana Dana We can fill out the rest of the tree What’s the Probability That Katherine Gets a ticket? Example Katherine Michael Katherine Dana Michael Katherine Michael Dana Dana Katherine Dana Michael

  23. A subjectiveprobability is a person’s estimate of the chance of an event occurring • This is based on personal judgment • Subjective probabilities should be between 0 and 1, but may not obey all the laws of probability • For example, 90% of the people consider themselves better than average drivers … Subjective Probability

  24. Probabilities describe the chances of events occurring … events consisting of outcomes in a sample space Probabilities must obey certain rules such as always being greater than or equal to 0 There are various ways to compute probabilities, including empirically, using classical methods, and by simulations Summary

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