Download Chapter 5: Discrete Probability Distributions

Chapter 5: Discrete Probability Distributions
Diana Pell
Section 5.1: Probability Distributions
A random variable is a variable whose values are determined by chance.
A discrete probability distribution consists of the values a random variable can assume and
the corresponding probabilities of the values. The probabilities are determined theoretically or
by observation.
Exercise 1. Construct a probability distribution for rolling a single die.
Exercise 2. Toss a coin three times and observe the number of heads. Find the probability
distribution and graph it.
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Exercise 3. The baseball World Series is played by the winner of the National League and that
of the American League. The first team to win four games wins the World Series. In other
words, the series will consist of four to seven games, depending on the individual victories. The
data shown consist of 40 World Series events. The number of games played in each series is
represented by the variable X. Find the probability P (X) for each X, construct a probability
distribution, and draw a graph for the data.
Two Requirements for a Probability Distribution
1. The sum of the probabilities of all the events in the sample space must equal 1
2. The probability of each event in the sample space must be between or equal to 0 and 1.
Exercise 4. Determine whether each distribution is a probability distribution.
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Section 5.2: Mean, Variance, Standard Deviation, and Expectation
Formula for the Mean of a Probability Distribution
The mean of a random variable with a discrete probability distribution is
µ = X1 · P (X1 ) + X2 · P (X2 ) + X3 · P (X3 ) + · · · + Xn · P (Xn ) =
X
X · P (X)
where X1 , X2 , X3 , . . . , Xn are the outcomes and P (X1 ), P (X2 ), P (X3 ), . . . , P (Xn ) are the corresponding probabilities.
Note: The rounding rule for the mean, variance, and standard deviation for variables of a probability distribution is this: The mean, variance, and standard deviation should be rounded to
one more decimal place than the outcome X. When fractions are used, they should be reduced
to lowest terms.
Exercise 5. Find the mean of the number of spots that appear when a die is tossed.
Exercise 6. In families with five children, find the mean number of children who will be girls.
Exercise 7. If three coins are tossed, find the mean of the number of heads that occur.
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Exercise 8. The probability distribution shown represents the number of trips of five nights or
more that American adults take per year. (That is, 6% do not take any trips lasting five nights
or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean.
Variance and Standard Deviation
Find the variance of a probability distribution by multiplying the square of each outcome by
its corresponding probability, summing those products, and subtracting the square of the mean.
The formula for the variance of a probability distribution is
X
σ2 =
[X 2 · P (X)] − µ2
Standard deviation of probability distribution is square root of the variance.
√
σ = σ2
Exercise 9. Compute the variance and standard deviation for the probability distribution in
Exercise 5.
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Exercise 10. A box contains 5 balls. Two are numbered 3, one is numbered 4, and two are
numbered 5. The balls are mixed and one is selected at random. After a ball is selected, its
number is recorded. Then it is replaced. If the experiment is repeated many times, find the
variance and standard deviation of the numbers on the balls.
Exercise 11. A talk radio station has four telephone lines. If the host is unable to talk (i.e.,
during a commercial) or is talking to a person, the other callers are placed on hold. When all
lines are in use, others who are trying to call in get a busy signal. The probability that 0, 1, 2,
3, or 4 people will get through is shown in the probability distribution. Find the variance and
standard deviation for the distribution.
Should the station have considered getting more phone lines installed?
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Expectation
The expected value of a discrete random variable of a probability distribution is the theoretical
average of the variable. The formula is
X
E(X) =
X · P (X)
Exercise 12. One thousand tickets are sold at $1 each for a color television valued at $350.
What is the expected value of the gain if you purchase one ticket?
Exercise 13. Six balls numbered 1, 2, 3, 5, 8, and 13 are placed in a box. A ball is selected
at random, and its number is recorded and then it is replaced. Find the expected value of the
numbers that will occur.
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Exercise 14. A financial adviser suggests that his client select one of two types of bonds in
which to invest $5000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a
2 12 % return and a default rate of 1%. Find the expected rate of return and decide which bond
would be a better investment. When the bond defaults, the investor loses all the investment.
Exercise 15. You pay $2 to play. You pick a card at random from a standard deck of cards. If
you pick the ace of spades, you win $13. Any other ace wins $10. Any face card (jack, queen, or
king) wins $5. All other cards win nothing. Do you want to play? Why or why not? Calculate
your expected winnings.
Note: If the expected value of the game is zero, the game is said to be fair. If the expected
value of a game is positive, then the game is in favor of the player. If the expected value of the
game is negative, then the game is said to be in favor of the house.
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Section 5.3: The Binomial Distribution
Experiments with just two outcomes are called binomial trials. Here are some examples of
binomial trials.
1. Toss a coin and observe the outcome, heads or tails.
2. Administer a drug to a sick individual and classify the reaction as “effective” or “ineffective.”
3. Manufacture a light bulb and classify it as “non-defective” or “defective.”
The outcomes of a binomial trial are usually called “success” or “failure.” We will denote the
probability of “success” by p and probability of “failure” by q. Since binomial trial has only two
outcomes we have p + q = 1, or
q =1−p
X - success
F - failure
p = P (X) and q = P (F ) and q = 1 − p
(1) Repeat experiment n times.
(2) Outcome is X or F .
(3) Repeated trials are independent.
What is the probability of k successes and n − k failures?
n k n−k
P (X = K) =
p q
k
Exercise 16. Find the probability of obtaining exactly two heads when tossing a fair coin three
times.
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Exercise 17. Find the probability of obtaining exactly 17 heads when tossing a coin 20 times.
Exercise 18. A plumbing-supplies manufacturer produces faucet washers, which are packaged
in boxes of 300. Quality control studies have shown that 2% of the washers are defective. What
is the probability that a box of washers contains exactly 9 defective washers?
Exercise 19. Each time at bat the probability that a baseball player gets a hit is .3. He comes
up to a bat four times in a game. Assume that his times at bat are independent trials. Find the
probability that he gets
a) exactly two hits
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b) at least two hits
Exercise 20. A survey found that 33% of people earning between 30, 000and75,000 said that
they were very happy. If 6 people who earn between 30, 000and75,000 are selected at random,
find the probability that at most 2 would consider themselves very happy.
Exercise 21. The recovery rate for a certain cattle disease is 25%. If 40 cattle are afflicted with
the disease, what is the probability that exactly 10 will recover?
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