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AP Statistics – Part IV: Randomness and Probability Chapter 17: Probability Models (page 386) • Bernoulli trials – (examples: tossing a coin, looking for defective products, shooting free throws) Outcomes are either PASS or FAILURE • Page 387: Calvin and Hobbes • Geometric Model: • We are more likely to want to know how long it will take us to achieve a success. The model that tells us this probability is called the Geometric Probability Model. Geometric models are completely specified by one parameter, p, the probability of success, and are denoted Geom(p). • Geometric Probability Model for Bernoulli trials: Geom(p) o p = probability of success (and q = 1 – p = probability of failure) o X = number of trails until the first success occurs o P(X=x) = qx-1 p o Expected Value: µ = 1 / p o Standard deviation: σ = √ (q / p2) • Trials must be Independent for Bernoulli trials. If you have an infinite supply of items (ex: the cereal boxes with Tiger Woods cards) then the fact that once you realize a card is NOT there, the change in probability would be insignificant. However, if you have a finite population, this does cause the probabilities to change, making the trials NOT INDEPENDENT. • The 10% Condition: If we look at less than 10% of the population, we can pretend that the trials are independent and still calculate probabilities quite accurately. o Bernoulli trials must be independent. If that assumption is violated, it is still OK to proceed as long as the sample is smaller than 10% of the population. • Page 389: Step-by-step • TI TIP: 2nd, Distr, geometpdf( is the “probability density function”. Allows you to find the probability of an individual outcome. Need p and x = the number of trials until you get a success. o geometpdf (p, x) • TI TIP: 2nd, Distr, geometcdf( is the “cumulative density function”. Finds the sum of the probabilities of several possible outcomes. Calculates the probability of finding the first success on or before the xth trial. o geometcdf (p, x) • Binomial Model: • Requires 2 parameters: o n = number of trials o p = probability of success. o Binom (n, p) • Each different order in which you can have k successes in n trials is called a combination. Total number of ways this can happen is n or nCk said, “n choose k”. k n! where n! = n x (n – 1) x … x 1 n = nC k = k k! (n – k)! • The probability of exactly k successes in n trials is n pkqn-k k o Expected Value: E(X) = np. o Standard Deviation: SD(X) = √npq • A Binomial probability model describes the number of successes in a specified number of trials. • Binomial Probability Model for Bernoulli trials: Binom (n, p) o N = number of trials o p = probability of success (and q = 1 – p = probability of failure) o X = umber of successes in n trials. n! o P(X = x) = n px qn-x, where n = x x x! (n – x)! o Mean: µ = np o Standard Deviation: σ = √npq • Page 392: Step-by-step • TI TIP: 2nd, Distr, binompdf( allows you to find the probability of an individual outcome. You need to define the Binomial model by specifying n and p, and then indicating the desired number of successes, x. o Binompdf(n,p,x) • TI TIP: 2nd, Distr, binomcdf( allows you to find the total probability of getting x or fewer successes among the n trials. o Binomcdf(n,p,x) • TI TIP: To find a “complement”, use 1 – binomcdf(n,p,x) • Binomial is discrete, giving probabilities for specific counts • Normal is continuous probability, so you cannot list all possible outcomes for the probabilities • Hints: o Be sure you have Bernoulli trials o Don’t confuse Geometric and Binomial models o Don’t use the Normal approximation with small n.