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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.

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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) • Did I mention MANY?

  4. Spinning Pennies • We will spin a penny 50 times… • What do you think the probability is of spinning a penny? • Write your guess in the blank on your half sheet. • Find a partner and run the activity while filling out the half sheet and graph (each person will spin 50 times). • OUR GOAL? Estimate the probability of spinning a head.

  5. The assignment was to SPIN a penny 50 times and record the number of heads and tails. As class we recorded _____ trials and the class average was _____. Why do you think this is?

  6. 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.

  7. 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

  8. 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.

  9. DO #’s 2,4

  10. #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?

  11. # 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.

  12. 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.

  13. 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.

  14. Work with your partner • DO #11,14 • Be complete and concise with your answers. • Do them NOW! 

  15. Probability Rules The probability of an event is written P(A). The complement, P(AC)= 1-P(A). Events A and B are disjoint or mutually exclusive if the have no outcomes in common. P(A or B)= P(A) + P(B) (addition rule) Events A and B are independent if P(A and B)= P(A)P(B).

  16. #11 • A) S = { germinates, fails to grow} • B) S = { 1,2,3,…..} depending on if you measure in days, weeks, months. • C) S = { A,B,C,D,F} • D) S = {HHHH, HHHM, HHMH, HMHH, MHHH, HHMM, HMHM, MMHH, MHMH, MHHM, HMMH, MMMH, MMHM, MHMM, HMMM, MMMM} • E) S = {4,3,2,1,0}

  17. #14 • A) 2 x 2 = 4 • S = {hh, ht, th, tt} • B) 2 x 2 x 2 = 8 • S = {hhh, hht, hth, thh, tth, tht, htt, ttt} • C) 2 x 2 x 2 x 2 = 16 • {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, TTHH, THTH, THHT, HTTH, TTTH, TTHT, THTT, HTTT, TTTT} • P(at least one head)=?

  18. Examples The probability of choosing each color of a peanut m&m is P(Blue)= P(Brown U Red)= P(Yellowc)=

  19. Homework • Read and Take notes 6.2 • Do #’s 3,5,12,15,18,19,21,23,26,30

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