probability - 4 Probability What is probability The...

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Page 4.1 (C:\data\StatPrimer\probability.wpd Print date: 8/1/06) 4: Probability What is probability? The probability of an event is its relative frequency (proportion) in the population. An event that happens half the time (such as a head showing up on the flip of a fair coin) has probability 50%. A horse that wins 1 in 4 races has a 25% probability of winning. A treatment that works in 4 of 5 patients has an 80% probability of success. Probability can occasionally be derived logically by counting the number of ways a thing can happen and determine its relative frequency. To use a familiar example, there are 52 cards in a deck, 4 of which are Kings. Therefore, the probability of randomly drawing a King = 4 ÷ 52 = .0769. Probability can be estimated through experience . If an event occurs x times out of n , then its probability will converge on X ÷ n as n becomes large. For example, if we flip a coin many times, we expect to see half the flips turn up heads. This experience is unreliable when n is small, but becomes increasingly reliable as n increases. For example, if a coin is flipped 10 times, there is no guarantee that we will observe exactly 5 heads. However, if the coin is flipped 1000 times, chances are better that the proportions of heads will be close to 0.50. Probability can be used to quantify subjective opinion. If a doctor says “you have a 50% chance of recovery,” the doctor believes that half of similar cases will recover in the long run. Presumably, this is based on knowledge, and not on a whim. The benefit of stating subjective probabilities is that they can be tested and modified according to experience. Notes: Range of possible probabilities: Probabilities can be no less than 0% and no more than 100% (of course). N otation: Let A represent an event. Then, Pr(A) represents the probability of the event. Complement: Let } represent the complement of event A. The complement of an event is its “opposite,” i.e., the event not happening. For example, if event A is recovery following treatment, then } represents failure to recover. Law of complements: Pr( } ) = 1 - Pr( A ). For example, if Pr( A ) = 0.75, then Pr( } ) = 1 ! 0.75 = 0.25. Random variable: A random variable is a quantity that varies depending on chance. There are two types of random variables, Discrete random variables can take on a finite number of possible outcomes. We study binomial random variables as examples of discrete random variables. Continuous random variables form an unbroken chain of possible outcomes, and can take on an infinite number of possibilities. We study Normal random variables as examples of continuous random variables.
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Page 4.2 (C:\data\StatPrimer\probability.wpd Print date: 8/1/06) (4.1) Binomial Distributions Binomial Random Variables Consider a random event that can take on only one of two possible outcomes. Each event is a Bernoulli trial . Arbitrarily, define one outcome a “success” and the other a “failure.” Now, take a series of n independent Bernoulli trials. The random number of successes in n Bernoulli trials is a binomial random variable.
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