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Probability

Probability. The branch of mathematics that describes the pattern of chance outcome. Chapter 6 Probability: The Study of Randomness. Probability calculations are the basis for inference. You can make predictions, describe trends, etc using probability.

cedric-adams
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Probability

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  1. Probability The branch of mathematics that describes the pattern of chance outcome.

  2. Chapter 6 Probability: The Study of Randomness • Probability calculations are the basis for inference. • You can make predictions, describe trends, etc using probability. • You ask the question “How often would this method give me the correct answer if I used it very many times?”

  3. BIG IDEA • Chance behavior is unpredictable in the short run BUT has a regular and predictable pattern in the long run. (MANY, MANY, MANY repetitions) • The assignment was to SPIN a penny 50 times and record the number of heads and tails. As class we recorded 145 trials and the class average was .4325. Why do you think this is?

  4. Warm Up! Woot woot! • On a clean sheet of paper, do problems #’s 11, 12, 15 • Compare answers with your group members and be ready to discuss as a class. • 10 minutes  GO!

  5. Chapter 6.1 problems • 6.2 • I did 20 simulations of randint(0,1,4) and recorded the number of times Betty won or loss {-4,-2,0,2,4}. • My simulation outcomes were 1/20 (-4), 6/20 (-2), 5/20 (0), 8/20 (2), 0/20 (4). • What were yours?

  6. 6.3 • A) Use your calculator to store SHAQ • B) He hit 47% of his shots for me. • How about you? • C) I had 5 Hits in a row and 5 Misses in a row. • How about you?

  7. 6.4 • A) 0 is the probability for an impossible event • B) 1 is the probability for an event that is certain. • C) .01 is the probability for an event that is very unlikely. • D) 0.6 is the probability that an event will occur more often than not.

  8. 6.5 There are 21 zeroes in the first 200 digits of the RDT. This seems very logical because its ratio has a proportion of .105 which is very close to 0.1. • 6.14 • 2x2 = 4 • 2x2x2 = 8 • 2x2x2x2 = 16

  9. 6.1 The Idea of Probability • What does it mean to say that a probability of a fair coin is one half, or that the chances I pass this class are 80 percent, or that the probability that the Panthers win the Super Bowl this season is .1? • A probability is a numerical measure of the likelihood of the event. It is a number that we attach to an event.

  10. We call a phenomenon RANDOM if individual outcomes are uncertain, but there is a regular distribution of outcomes in a larger number of repetitions. • Probability Of An Event P(A) =  The Number Of Ways Event A Can Occur    The Total Number Of Possible Outcomes

  11. Randomness • You must have a long series of independent trials. (one outcome does not influence the outcome of any other) • The idea of probability is empirical. (based on observation rather than theory) • Short runs only give estimates, computer simulations are very useful so to be able to do LONG RUN of simulations.

  12. Where do probabilities come from? • Probabilities may be given, often in the form of a table. For example, if an experiment has three possible outcomes: Apple, Banana, and Cherry, one might be given the following table at the right : • Probabilities my be historical, if it has rained during 1/3 of the days in June during the past, one may say that the probability of rain for a day in June is 1/3. • Probabilities may be theoretical, if a die is fair, since there are six possible outcomes; the probability of getting a 3 is 1/6.

  13. Problem: A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble? • Outcomes: The possible outcomes of this experiment are red, green, blue and yellow. • Probabilities: • P (RED) = • P (GREEN) = • P (BLUE) = • P (YELLOW) =

  14. Conclusion: The outcomes in this experiment are not equally likely to occur. You are more likely to choose a blue marble than any other color. You are least likely to choose a yellow marble.

  15. 6.2 Probability Models

  16. The sample space S of a random phenomenon is the set of all possible outcomes. • Example: If the experiment is to throw a standard die and record the outcome, the sample space is S = {1, 2, 3, 4, 5, 6}, the set of possible outcomes.

  17. An event is any outcome or a set of outcomes of a random phenomenon. • An event is a subset of the sample space. • A probability model is a mathematical description of a random phenomenon. • Consists of two parts: a sample space, S and a way of assigning probabilities to events.

  18. Techniques for finding the number of outcomes: • Tree Diagram – represent the first action, then draw “branches” to the next set of actions. • Multiplication Principle (of Counting) – do one task in “a” number of ways and another task in “b” number of ways, then both tasks can be done in “a” x “b” ways.

  19. Examples: • If you have 4 sweaters, 3 shirts, and 5 pairs of pants, how many different outfits can you make using all three types of clothes? • 4 x 3 x 5 = 60 • How many total possible outcomes are there from tossing a coin, rolling a die and then tossing a coin? • 2 x 6 x 2 = 24, can you list the sample space?

  20. Discrete vs. Continuous • If the experiment is to throw a standard die and record the outcome, the sample space is: • S = {1, 2, 3, 4, 5, 6}, the set of possible outcomes. • Discrete sample space • Finite number of outcomes • On the other hand, if the experiment is to randomly pick a number between 0 and 1, then the sample space is to be S = [all #’s between 0 and 1], • Continuous sample space • Infinite number of outcomes

  21. Replacement v. Non-replacement • If you are selecting objects from a finite group of objects, whether you replace the object is very important. • If you do not replace, then the probability for each selection will change. • If we sample with replacement, each item is replaced in the population before the next draw; thus, a single object may occur several times in the sample. • If we sample without replacement, objects are not replaced in the population.

  22. Example: 3 marbles are drawn from a jar of 4 yellow, 5 blue, and 2 red. What is the probability of getting 3 blue (in other words, you draw thrice ) • With replacement • Without replacement

  23. Example: • What is the total possible outcomes in the sample space of the NC Pick Three? • What is the probability that you have a winning ticket that is a 2 2 5?

  24. #1 Any probability is a number from 0 to 1. An event that never occurs has a probability of zero. An event that occurs on every trial has a probability of 1. An event with a probability of .5 occurs in half the trials in the long run. Probability Rules

  25. #2 All possible outcomes together must have the probability of 1. ( there sum = 1) • #3 The probability that an event does not occur is 1 minus the probability that the event does occur. • #4 If two events have no outcomes in common, the probability that one or the other occurs is the sum of the individual probabilities.

  26. Homework • Finish Reading and Taking Notes on 6.2 p335-359 do #’s 18,19,21,23,26,30 • Read and Take notes on Chapter 4.3 p241-255 do #’s 50, 51, 52, 53

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