Chapter 4 Distributions

Transcription

1 Chapter 4 Distributions Probability Distributions Symbols: Conditions: Prob. is between P(x) = Sum of all prob. is Decide whether the random variable x is discrete or continuous: 1. x represents the length of time it takes to complete a test 2. x represents the number of home runs hit during a baseball game A company tracts the number of sales new employees make each day during a 100-day probationary period. The results for one employee are show on the left. 3. Construct a probability distribution. Sales per Number of day, x days, f X P(x) Ms. Ellis family likes to gamble on the college football bowl games. There are 24 family members that participate. Each family member bets $10 and fills out a football bowl game form. The winner gets $50. The second place finisher gets $25. The rest of the money gets donated to charity (No one wins it!). 4. What is the expected value for Ms. Ellis to win the family bowl game bet?

2 Binomial Distributions Symbols: n = Criteria: 1. Fixed number of independent trials p = 2. Outcomes: success or failure q = 3. Prob. success equal for every trial 4. X counts number of successes Decide whether the following experiment is a binomial experiment. If it is, identify a success, specify the values of n, p, q, and list the possible values of the random variable x. 5. From past records, a clothing store finds that 26% of the people who enter the store will make a purchase. During a one-hour period, 18 people enter the store. The random variable x represents the number of people who do NOT make a purchase. Formulas: mean = Calc. Help: binomialpdf(,, ) Var = binomialcdf(,, ) S.D. = Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p. 6. n = 100, p = n = 342, p = 0.63 Fifty-four percent of men consider themselves basketball fans. You randomly select 10 men and ask each if he considers himself a basketball fan. Find the probability that the number of men who consider themselves fans is: 8. P(exactly 8) 9. P(8 or less) 10. P(less than 8) 11. P(at least 8)

3 Geometric Distribution Symbols: p = Criteria: 1. Repeat until success occurs q = 2. Trials independent 3. Prob. success same for every trial 4. Random variable x represents number of trials in which success occur Calc Help: Geometpdf(, ) Geometcdf(, ) Assume the probability that you will make a sale on any given phone call is Find the probability that you: 12. Make your first sale on the fifth call 13. Make your first sale on the first, second, third or fourth call 15. Do not make a sale on the first three calls A cereal maker places a game piece in its cereal boxes. The probability of winning a prize in the game is 1 in 4. Find the probability that you: 16. Win your first prize with your fourth purchase 17. Win your first prize with your first, second or third purchase 18. Do not win a prize in the first four purchases

4 Poisson Distribution Symbols: µ = Criteria: Gives prob. of x occurrences over a specified interval of time, area or volume. Calc Help: poissonpdf(, ) Poissoncdf(, ) A study on Ms. Ellis typing skills find that the mean number o typographical errors per page is 4. Find the probability that: 19. exactly three typographical errors will be found on a page 20. at most three typographical errors will be found on a page 21. at most five typographical errors will be found on a page It is estimated that shark attacks kill 10 people each year worldwide. Find the probability that sharks kill at least three people this year: 22. assuming that this rate is true. 23. if the rate is actually five people a year. 24. if the rate is actually 15 people a year.

5 YOU PICK!! What distribution is it? 1. A farmer is worried about a drought. The Farmer s Almanac records an average of 7 days of rain throughout the month of May. He knows that if he doesn t get more than 3 days of rain his seeds will not germinate. B. What is the probability that he will get exactly 4 days of rain? C. What is the probability that his crops will germinate? D. What is the probability that he will receive 7 days of rain or less? 2. The McDonald s Monopoly game is designed to have one winner for every four players. B. What is the probability that you will win on your first play? C. What is the probability that you will not win until your 6 th purchase? D. What is the probability that you will win by your 6 th purchase? E. What is the probability you will not win in your first four purchases? 3. The FAA keeps track of the percentage of on time flights for each airline. Recently, US Airways was on time for 78% of its flights. A US sales manager knows she will be taking 6 US Airways flights next month for business. B. Form a probability distribution for x. (x represents an on time flight) C. What is the probability that exactly 3 flights will be on time? D. What is the probability that 5 or more flights will be on time? E. By using standard deviation, find the range for the number of expected flights that lie within two standard deviations of the mean.