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Ch16 Binomial vs. Geometric Distributions
Binomial: According to the M&M’s website the percent of blue M&M’s in each bag is 24%. You select a sample of 10 M&Ms
(without looking) from a bag. Let X = the number of blue M&Ms.
Geometric: According to the M&M’s website the percent of brown M&M’s in each bag is 13%. Select an M&M (without looking)
from a bag until a brown is drawn. Let X = the number of M&Ms until a brown is drawn.
For each of the following situations, determine if the situation is binomial, geometric, or neither. If neither explain why.
1.
You observe the sex of the next 20 children born at a local hospital.
a) X is the number of females among them.
b) X is the number of children you observe until you see the first female
2.
A couple decides to continue to have children until their first girl is born: X is the total number of children the couple has.
3.
Bobby draws cards without replacement from a deck until he gets an ace. He then reshuffles, and starts over until he “wins” 10
times. The count X is the total number of cards he counted until he finished.
4.
Alexandra spins a coin 5 times each day for a year.
a) X is the number of coins until she gets the first tail
b) X is the number of tails she observes each day.
5.
A student studies statistics using computer-assisted instruction. After the lesson, the computer presents 10 problems. The student
solves each problem and enters her answer. The computer gives additional instruction between problems if the answer is wrong.
The count X is the number of problems that the student gets right.
6.
Sam buys a “Texas 2-Step” lottery ticket every week. The count X is the number of times in a year that he wins a prize.
7.
Assume that 13% of people are left-handed and we select 5 people at random. Identify the following as Binomial or Geometric.
Do not solve!
a) The first lefty is the fifth person chose.
b) There are some lefties among the 5 people.
c) The first lefty is the second or third person.
d) There are exactly 3 lefties in the group.
e) There are at least 3 lefties in the group.
f) There are no more than 3 lefties in the group.
g) How many lefties do you expect in this group?
h) If we keep picking people until we find a lefty, how long do you expect it will take?
Binomial Distributions WS
If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X and 0,
1, 2, ... , n. If k is any one of these values, then:
nI
b gF
G
Hk J
Kp b1  pg
P X k 
k
n k
Example 1: A fair coin is flipped at the beginning of each football game to determine which team can opt to receive or kick the
football. If a professional football team plays 10 games in the regular season, what is the probability that:
a) the team will have their option to kick or receive exactly 4 times?
b) the team will have at least 8 options?
c) the team will have at most 3 options?
If X is a random variable with a binomial distribution, then
Mean Value (Expected Value):
E ( X )   x  np
Standard Deviation:
 x  np1  p
Example 2: The number of inaccurate gauges (defects) in a group of four is a binomial random variable. If the probability of a defect
is 0.1, a) what is the probability that only 1 is defective?
b) More than 1?
c) Determine the probability distribution for the number of inaccurate gauges.
d) What is the expected number and the standard deviation of inaccurate gauges?
Example 3: A certain medical test is known to detect 90% of the people who are afflicted with disease Y. If 8 people with the disease
are administered the test, what is the probability that the test will show that:
a) all 8 have the disease?
b) at least 3 people have the disease?
c) at most 2 have the disease?
Homework: Work it on a separate sheet of paper.
1. Let X = the count described. Does X have a binomial distribution? Give your reasons in each case.
a. We roll 50 dice to find the distribution of the number of spots on the faces.
b. How likely is it that in a group of 120 the majority may have Type A blood, given that Type A is found in 43% of the population?
c. We deal 5 cards from a deck and get all hearts. How likely is it?
d. We wish to predict the outcome of a vote on the school budget, and poll 500 of the 3000 likely voters to see how many favor the
proposed budget.
e. A company realizes that about 10% of its packages are not being sealed properly. In a case of 24, is it likely that more than 3 are
unsealed.
2. Each year a company selects a number of employees for a management training program. On average, 70% of those sent complete
the program. Out of the seven people sent, what is the probability that
a. Exactly five complete the program?
b. Five or more complete the program?
3. A manufacturing process produces 3 % defective items. The company ships 12 items in each box and wishes to guarantee no more
than one defective item per box. What is the probability that the box will fail to satisfy the guarantee?
4. Suppose it is known that 80% of the people exposed to the flu virus will contract the flu. Out of a family of five exposed to the
virus, what is the probability that:
a. No one will contract the flu?
b. All will contract the flu?
c. Exactly two will get the flu?
d. At least two will get the flu?
e. Let X = number of family members contracting the flu. Complete a probability distribution of X.
f. Find E(X) and Var(X)
5. A person with tuberculosis is given a chest x-ray. Four TB x-ray specialists examined each x-ray independently. If each specialist
can detect TB 88% of the time when it is present, what is the probability that less than three specialists will detect the presence of TB?