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UCSB ECON 240a - Probability

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Sept. 29, 2005 LEC #3 ECON 240A-1 L. PhillipsProbabilityI. IntroductionProbability 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 thetheory 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 randomsample 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 beapproximated 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 ExperimentsA 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.Sept. 29, 2005 LEC #3 ECON 240A-2 L. PhillipsProbability------------------------------------------------------------------------------------------------------------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.HT123456Figure 1: Flipping a Coin Figure 2: Rolling a DieSept. 29, 2005 LEC #3 ECON 240A-3 L. PhillipsProbabilityProbabilities of elementary outcomes are non-negative and the probabilities of all the elementary outcomes sum to one.III. EventsElementary outcomes can be grouped into sets that define an event. For example, in the random experiment of throwing a die, the die can come up even, with the set of elementary outcomes {2,4,6}, or the die can come up odd with the set of elementary outcomes {1, 3, 5}. The probability of the event even is the sum of the probabilities of theelementary outcomes in its set, so if the die is fair, P(even) = P(2) + P(4) + P(6) = 3/6.Consider the random experiment of flipping a nickel and a quarter simultaneously.There are four elementary outcomes: (1) nickel heads and quarter heads, {NH, QH}, (2) nickel heads and quarter tails, {NH, QT}, (3) nickel tails and quarter heads, {NT, QH}, and (4) nickel tails and quarter tails, {NT, QT}.This is illustrated in Figure 3.------------------------------------------------------------------------------------------------------------NH, QH NH, QTNT, QH NT, QTFigure 3: Flipping a Nickel and a Quarter Simultaneously, Elementary Outcomes------------------------------------------------------------------------------------------------------------Once again, these elementary events can be combined into events, for example, the event nickel heads: event {Nickel Heads} = {{NH, QH} {NH, QT}}. Assuming the coins are fair, the probability of the event nickel heads, P (NH), is the sum of the probabilities of the elementary events that compose it: P(NH) = P(NH, QH) + P(NH, QT) = 2/4.Sept. 29, 2005 LEC #3 ECON 240A-4 L. PhillipsProbabilitySimilarly, the event quarter tails involves two elementary outcomes: event {QT} = {{NH, QT} {NT, QT}}, with probability,P(QT) = P(NH, QT) + P(NT, QT) = 2/4.These events, nickel heads and quarter tails can also be combined into events: (1) nickel heads or quarter tails, {nickel headsquarter tails}, i.e. one event or the other takesplace, (2) nickel heads and quarter tails, {nickel headsquarter tails}, i.e. both events takeplace. (3)The event not nickel heads, {NH }, called the event complementary to nickel heads, is equivalent to the event nickel tails, {NT}.IV. The Addition RuleThe event {nickel heads or quarter tails}, {NHQT}, includes three elementary events:{NHQT} = {NH,QH}, {NH,QT}, {NT,QT}. The probability of nickel heads or quarter tails is the probability of the first row of Figure 3, plus the probability of the second column of Figure 3, minus the overlap or intersection, i.e. the double counting of the upper right hand corner. This leads to the addition rule: P(NHQT) = P(NH) + P(QT) – P(NHQT) = 2/4 +2/4 – ¼ = ¾.If two events, e.g. event A and event B are mutually exclusive, then P(AB) = 0, so P() = P(A) + P(B). Since elementary events are mutually exclusive by definition, events composed of combining them, for example nickel heads (above), have probabilities which are just the sum of the elementary outcomes composing this event, e.g. recall that P(NH) = P(NH, QH) + P(NH, QT) = 2/4. The probability of an event, and the probability of not the event must sum to one, for example, P(NH) + P(NH) =1.Sept. 29, 2005 LEC #3 ECON 240A-5 L. PhillipsProbabilityV. Interpretations or Meanings of Probability There are various perspectives on the meaning of probability. One meaning is classical or a perspective based on gambling. The basic assumption is that the gamble or random experiment is fair and that the elementary outcomes are equally likely.But not all uncertainty in life is associated with gambles. Much of life itself is uncertain. For example, will the Dow Jones Industrials end the fourth quarter of the year 2005 higher than it was at the end of the third quarter? One approach to such questions is empirical. In the last ten years, what fraction of the quarters witnessed a rise in the Dow?Another perspective on uncertainty in events is subjective or personal. For example,the likelihood of success in your academic career and professional career could be


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UCSB ECON 240a - Probability

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