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MATH 221

MATH 221. Integrated Learning System Week 3, Lecture 1 More Discrete Probability Distributions. Objectives:. At the end of this presentation you should be able to find probabilities using the geometric distribution and the Poisson distribution. Definition.

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MATH 221

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  1. MATH 221 Integrated Learning System Week 3, Lecture 1 More Discrete Probability Distributions

  2. Objectives: At the end of this presentation you should be able to find probabilities using the geometric distribution and the Poisson distribution.

  3. Definition • A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions: • The trial is repeated until a success occurs. • The repeated trials are independent. • The probability of success p is constant for each trial.

  4. Geometric Probability Function P(x) = pqx-1where q = 1 – p.

  5. Geometric Probability Example • A cereal maker places a game piece in its cereal boxes. The probability of winning a prize in the game is ¼. Find the probability that you: • Win your first prize with your 4th purchase. • Win your first prize with your 1st, 2nd, or 3rd purchase. • You do not win a prize with your first four purchases. • (Problem 10, page 193, Larson and Farber)

  6. Geometric Distribution So, what does the distribution we have just used look like?

  7. Using Technology: TI-83 geometpdf(p,x) – Probability density function geometcdf(p,x) – Cumulative probability density function

  8. Using Technology: EXCEL While EXCEL contains many of the probability density functions that we will use in this course, it does not have a geometric probability density function. However, the geometric probability density function is fairly simple and it is easy to enter as a cell function.

  9. Exercise • Basketball Player Shaquille O’Neal makes a free throw shot about 53.4% of the time. Find the probability that: • The first shot O’Neal makes is the second shot. • The first shot O’Neal makes is the first or second shot. • O’Neal does not make two shots.

  10. Definition • The Poisson distribution is a discrete probability distribution that satisfies the following • The experiment consists of counting the number of times x that an event occurs in a given interval. • The probability of the event is the same for each interval. • The number of occurrences is independent from interval to interval.

  11. Poisson Probability Function In the versions of the function above  and  represent the mean arrival rate or rate of occurrence for the event of interest.  is the lower case Greek letter mu,  is the lower case Greek letter lambda, and e is the base of the natural logarithms.

  12. Example • A newspaper finds that the mean number of typographical errors per page is four. Find the probability that: • Exactly 3 typographical errors will be found on a page. • At most 3 typographical errors will be found on a page. • More than 3 typographical errors will be found on a page.

  13. Example (Solution)

  14. Using Technology: TI-83 poissonpdf(,x) – Probability density function poissoncdf(,x) – Cumulative probability density function

  15. Using Technology: EXCEL

  16. Using Technology: EXCEL

  17. Using Technology: EXCEL

  18. Exercise • A major hurricane is a hurricane with wind speeds of 111 mph or greater. From 1900 to 1999, the mean number of major hurricanes to strike the U.S. mainland per year was about 0.6. Find the probability that in a given year: • Exactly one major hurricane will strike the U.S. • At most one major hurricane will strike the U.S. • More than one major hurricane will strike the U.S.

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