More Discrete Probability Distributions 4.3 216 CHAPTER 4 Discrete Probability Distributions The Geometric Distribution The Poisson Distribution Summary of Discrete Probability Distributions What You Should Learn How to find probabilities using the geometric distribution How to find probabilities using the Poisson distribution The Geometric Distribution Many actions in life are repeated until a success occurs. For instance, you might have to call a number several times before someone answers. A situation such as this can be represented by a geometric distribution. A geometric distribution is a discrete probability distribution of a random variable x that satisfies these conditions. 1. A trial is repeated until a success occurs. 2. The repeated trials are independent of each other. 3. The probability of success p is the same for each trial. 4. The random variable x represents the number of the trial in which the first success occurs. The probability that the first success will occur on trial number x is P1x2 = pqx-1, where q = 1 - p. DEFINITION For instance, when the first success occurs on the third trial, the outcome is FFS, and the probability is P132 = q# q# p, or P132 = p# q2. Using the Geometric Distribution A recent study found that the failure rate of businesses after five years is 50%. Four businesses that started five years ago are selected at random. Find the probability that the fourth business selected is the first one to have failed. (Source: U.S. Bureau of Labor Statistics) SOLUTION Using p = 0.50, q = 0.50, and x = 4, you have P142 = 0.5010.5024-1 = 10.5024 ≈ 0.063. So, the probability that the fourth business selected is the first one to have failed is about 0.063. You can use technology to check this result. For instance, using a TI-84 Plus, you can find P142, as shown below. TI-84 PLUS geometpdf(.5,4) .0625 EXAMPLE 1 Tech Tip You can use technology such as Minitab, Excel, StatCrunch, or the TI-84 Plus to find a geometric probability. For instance, here are instructions for finding a geometric probability on a TI-84 Plus. From the DISTR menu, choose the geometpdf( feature. Enter the values of p and x. Then calculate the probability.
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