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4.5: Geometric Distributions

CHS Statistics. 4.5: Geometric Distributions. Objective : To solve multistep probability tasks with the concept of geometric distributions. Geometric Distributions.

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4.5: Geometric Distributions

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  1. CHS Statistics 4.5: Geometric Distributions Objective: To solve multistep probability tasks with the concept of geometric distributions

  2. Geometric Distributions • A Geometric probability modeltells us the probability for a random variable that counts the number of trials until the first success.

  3. Geometric Distributions Requirements of geometric distributions: • Each observation is in one of two categories: success or failure. • The probability is the same for each observation. • Observations are independent. (Knowing the result of one observation tells you nothing about the other observations.) • The variable of interest is the number of trials required to obtain the first success. [The only difference from a binomial distribution]

  4. Example Does this represent a geometric distribution? What is your evidence? • A new sales gimmick is to sell bags of candy that have 30% of M&M’s covered with speckles. These “groovy” candies are mixed randomly with the normal candies as they are put into the bags for distribution and sale. You buy a bag and remove candies one at a time looking for the speckles.

  5. Geometric Model A new sales gimmick is to sell bags of candy that have 30% of M&M’s covered with speckles. These “groovy” candies are mixed randomly with the normal candies as they are put into the bags for distribution and sale. You buy a bag and remove candies one at a time looking for the speckles. • What’s the probability that the first speckled one we see is the fourth candy we get? Note that the skills to answer this question come from the very first day of the probability unit.

  6. Geometric Model (cont.) • What’s the probability that the first speckled one is the tenth one? Write a general formula. • What’s the probability that the first speckled candy is one of the first three we look at? • How many do we expect to have to check, on average, to find a speckled one?

  7. Geometric Model (cont.) p = probability of success q = 1 – p = probability of failure X = number of trials until the first success occurs P(X = x) = qx-1p

  8. Example • People with O-negative blood are “universal donors.” Only about 6% of people have O-negative blood. • If donors line up at random for a blood drive, how many do you expect to examine before you find someone who has O-negative blood? • What’s the probability that the first O-negative donor found is one of the four people in line?

  9. Geometric Probabilities Using Calculator • 2nd DISTR  geometpdf( • Note the pdf for Probability Density Function • Used to find any individual outcome • Format: geometpdf(p,x) • 2nd DISTR  geometcdf( • Note the cdf for Cumulative Density Function • Used to find the first success on or before the xth trial • Format: geometcdf(p,x) • Try the last example using the calculator! Much easier…

  10. Example Example: Let x represent the number of students who must be stopped before finding one with jumper cables. Suppose 40% of students who drive to school carry cables. Find the probability that the • 3rdperson you stop has them. • You need to stop no more than 3 people.

  11. Assignment

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