lecturethree - Sept 30 2010 LEC#3 I Introduction ECON...

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Sept. 30, 2010 LEC #3 ECON 140A/240A- 1 L. Phillips Probability I. Introduction Probability has its origins in curiosity about the laws governing gambles. During the Renaissance, the Chevalier De Mere posed the following puzzle. Which is more likely (1) rolling at least one six in four throws of a single die or (2) rolling at least one double six in 24 throws of a pair of dice? De Mere asked his friend Blaise Pascal this question. In turn Pascal enlisted the interest of Pierre De Fermat. Pascal and Fermat worked out the theory of probability. While gambling and probability are interesting topics in their own right, probability is also a step towards making statistical inferences. The key is using a random sample combined with probability models to estimate, for example, the fraction of voters who will vote for a candidate. The pathway of understanding is a sequence of Bernoulli trials leading to the binomial distribution, which for large numbers of observations can be approximated by the normal distribution. These distributions can be used to estimate the fraction that will vote yes and to calculate intervals within which the true fraction will lie say 95 percent of the time. II. Random Experiments A key to understanding probability is to model certain activities such as flipping a coin or throwing a die. The set of elementary outcomes, e.g. {heads, tails} or symbolically {H, T}, is the sample space. Branching or tree diagrams illustrate these random experiments and their elementary outcomes.
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Sept. 30, 2010 LEC #3 ECON 140A/240A- 2 L. Phillips Probability ----------------------------------------------------------------------------------------------------------- - Flipping a coin is an example of a Bernoulli trial, a random experiment with two elementary outcomes, such as yes/no or heads/tails. In a fair game, the probability of heads equals that of tails, for example, but the laws of probability hold whether the game is fair or not. If the game is fair, the probabilities of elementary outcomes for simple random experiments are intuitively obvious, i.e. equally likely. In the case of flipping a coin, the probability of heads, P (H), equals the probability of tails, P (T), equals one half. In the example of rolling a die, the probability of rolling a one, P (1), equals the probability of rolling a six, P (6), equals one sixth. H T 1 2 3 4 5 6 Figure 1: Flipping a Coin Figure 2: Rolling a Die
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Sept. 30, 2010 LEC #3 ECON 140A/240A- 3 L. Phillips Probability Probabilities of elementary outcomes are non-negative and the probabilities of all the elementary outcomes sum to one.
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