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Introduction to Probability

Introduction to Probability. Presentation 4.1. Chance. The idea of chance is everywhere What is the chance that I just bought the winning lottery ticket? What is the chance that I will be struck by lightning? it’s nearly double the chance of purchasing that winning lottery ticket

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Introduction to Probability

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  1. Introduction to Probability Presentation 4.1

  2. Chance • The idea of chance is everywhere • What is the chance that I just bought the winning lottery ticket? • What is the chance that I will be struck by lightning? • it’s nearly double the chance of purchasing that winning lottery ticket • What is the chance of that I will purchase that product after seeing the commercial? • What is the chance that a woman will be elected President in my lifetime? • What is the chance that you will live to be 100?

  3. Definition of Probability The definition of probability is: What is the probability of rolling a 2 on a 6-sided die? What is the probability of tossing a head on a coin?

  4. The short and Long of Probability • In the short run, chance behavior is quite unpredictable. • Consider tossing the coin, it would not be that unusual for the first four or five tosses to come up heads. • Consider tossing the coin once, using that result, the proportion of tosses (just the one) that comes up heads is either 1 (100%) or 0 (0%). • In the long run, chance behavior is very regular and predictable.

  5. The short and Long of Probability • As you continue to toss the coin more and more, the proportion of heads that come up should slowly approach 0.5 (or 50%). • The website below shows this in action. • As you toss the coin quickly 10 times, watch the percentage of heads. Do this one several times. • Then, toss the coin quickly 100 times and watch the percentage of heads. • Finally, toss the coin quickly 1000 times and watch the percentage of heads. • Did the percentage does indeed settle in toward 50% as the number of tosses increases? • Proceed to: http://nlvm.usu.edu/en/nav/frames_asid_305_g_3_t_5.html to answer the above questions.

  6. Over the Long Run • This is sometimes known as the Law of Large Numbers. • As the number of trials increases, the empirical probability (the probability in real trials) approaches the theoretical probability). • For the coin tossing, the empirical probability of tossing a head may be 0.7 (if you had tossed 7 heads in 10 tosses), while the theoretical would always be 0.5. • The graph below shows how the empirical probability “settles in” towards the theoretical as you toss the coin more and more.

  7. Independence • An assumption (and a rather safe one) that we are working under is that the tosses of the coin are independent. • That is, each toss is completely separate and its own event (with a probability of a head being 0.5) • For example, if you toss the coin five times and it comes up tails five times, what is the probability that it comes up tails on the next flip. • It is still (and always) 0.5. • The coin simply does not remember how it flipped the last time!

  8. Introduction to Probability • This concludes this presentation.

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