1. Discrete Probability Distributions

2. Probability Distributions

3.  A random variable x represents a numerical value associated with each outcome of a probability experiment.  It is DISCRETE if it has a finite number of possible outcomes.  It is CONTINUOUS if it has an uncountable number of possible outcomes (represented by an interval)

4.  13. the number of books in a university library.  19. the amount of snow (in inches) that fell in Nome, Alaska last winter.

5.  - list of each possible value and its probability. Must satisfy 2 conditions:  1. 0 < P(x) < 1  2. Σ P(x) = 1

6.  28. the # of games played in the World Series from 1903 to 2009 # of games played 45678 Frequency2023 363

7.  MEANµ = Σ [x·P(x)]  VARIANCE σ 2 = Σ [(x - µ) 2 ·P(x)]  STANDARD DEVIATION σ = √ σ 2

8.  36. The # of 911 calls received per hour. X01234567 P(x)0.10.100.260.330.180.060.03

9. Notation: E(x) Expected value represents what you would expect to happen over thousands of trials. SAME as the MEAN!!! E(x) = µ = Σ [x·P(x)]

10.  If x is the net gain to a player in a game of chance, then E(X) is usually negative. This value gives the average amount per game the player can expect to lose.  46. A charity organization is selling $5 raffle tickets. First prize is a trip to Mexico valued at $3450, second prize is a spa package valued at $750. The remaining 20 prizes are $25 gas cards. The number of tickets sold is 6000.

11. Binomial Distributions

12.  CONDITIONS:  1. there are a fixed number of independent trials (n = # of trials)  2. Two possible outcomes for each trial, Success or Failure.  3. Probability of Success is the same for each trial. p = P(Success) and q = P(Failure)  4. random variable x = # of successful trials

13. If binomial, ID ‘success’, find n, p, q; list possible values of x. If not binomial, explain why.  10. From past records, a clothing store finds that 26% of people who enter the store will make a purchase. During a one-hour period, 18 people enter the store. The random variable represents the # of people who do NOT make a purchase.

14.  To find the probability of (exactly) x number of successful trials:  P(x) = n C x · p x · q n –x

15.  18. A surgical technique is performed on 7 patients. You are told there is a 70% chance of success. Find the probability that the surgery is successful for  A) exactly 5 patients  B) at least 5 patients  C) less than 5 patients

16.  MEANµ = np  VARIANCE σ 2 = npq  STANDARD DEVIATION σ = √ σ 2

17.  Construct a probability distribution, then find mean, variance, and standard deviation for the following:  28. One in four adults claims to have no trouble sleeping at night. You randomly select 5 adults and ask them if they have trouble sleeping at night.