1. Discrete Probability Distributions
chAPTER four
Discrete Probability Distributions

2. Section 4.1
Probability Distributions

3. Random Variables
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. EX: Discrete or Continuous?
The number of books in a university library.
The amount of snow (in inches) that fell in Nome, Alaska last winter.

5. Discrete Probability Distributions
A list of each possible value and its probability. Must satisfy 2 conditions:
< P(x) < 1
2. ΣP(x) = 1

6. EX: make a probability distribution
The # of games played in the World Series from 1903 to 2009
# of games played 4 5 6 7 8
Frequency 20 23 36 3

7. To Find…

8. Find µ, σ2, and σ The # of 911 calls received per hour. X 1 2 3 4 5 61 2 3 4 5 6 7
P(x)
0.1
0.10
0.26
0.33
0.18
0.06
0.03

9. Expected Value
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. EX: find expected NET GAIN
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.
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. Section 4.2
Binomial Distributions

12. Binomial Experiments 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 binomial, ID ‘success’, find n, p, q; list possible values of x. If not binomial, explain why. 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. Binomial Probability Formula
To find the probability of exactly x number of successful trials:
P(x) = nCx · px · qn –x

15. EX: Find the indicated probability
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. To Find…

17. Construct a probability distribution, then find mean, variance, and standard deviation for the following:
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.
Consider “success” to mean no trouble sleeping.