Discrete Probability

Distributions

Probability Distributions

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)

13. the number of books in a university library.

19. the amount of snow (in inches) that fell in Nome, Alaska last winter.

- list of each possible value and its probability. Must satisfy 2 conditions:

1. 0 < P(x) < 1 2. ΣP(x) = 1

28. the # of games played in the World Series from 1903 to 2009 # of

games played

4 5 6 7 8

Frequency

20

23

23

36

3

MEAN µ = Σ[x·P(x)]

VARIANCE σ2 = Σ[(x - µ)2·P(x)]

STANDARD DEVIATION σ = √σ 2

36. The # of 911 calls received per hour.

X 0 1 2 3 4 5 6 7

P(x)

0.1 0.10

0.26

0.33

0.18

0.06

0.03

0.03

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)]

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.

Binomial Distributions

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

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.

To find the probability of (exactly) x number of successful trials:

P(x) = nCx · px · qn –x

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

MEAN µ = np

VARIANCE σ 2 = npq

STANDARD DEVIATION σ = √σ 2

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.