Download Ch16 Geometric WS and KEY

Ch16 Geometric Random Variable
Properties
1. Only two mutually exclusive outcomes (success or failure).
2. Independent trials.
3. Probability of outcome is constant for all trials, p.
4. Variable of interest is the number of trials required to get the first success.
Probability Formula
P( X  a)  1  p 
a 1
p
X is geometric random variable where a is the number of attempts to a success.
If X is a random variable with a geometric distribution, then
Mean Value (Expected Value):
E( X )   x 
1
p
Standard Deviation:
x 
1 p
p2
Example 1: The state department is trying to identify an individual who speaks Farsi to fill a foreign embassy position. They have
determined that 4% of the applicant pool is fluent in Farsi.
a) Find the mean and standard deviation of the number of people they must interview until they find the first one who is fluent in Farsi?
b) What is the probability that they will have to interview 3 until they find one who speaks Farsi?
c) What is the probability that they will have to interview no more than 5 until they find one who speaks Farsi?
d) What is the probability that they will have to interview more than 2 until they find one who speaks Farsi?
Example 2: A basketball player makes 80% of her free throws. We put her on the free throw line and ask her to shoot free throws until
she misses one. Let X = the number of free throws the player takes until she misses. (Assuming the shots are independent)
a) What is the probability that she misses for the first time on her 6 throw?
b) What is the probability that she will make at most 5 shots before she misses?
Homework:
Write on a separate sheet of paper.
1. Identify each of the following examples are Binomial, Geometric or neither.
a) Toss 3 fair nickels.
b) Roll a fair six sided die5 times and X is the number of “2’s”
c) Roll two fair six sided dice ten times.
d) Randomly sample 100 US adults and X is the number of males.
e) Flip a coin until you get a head.
f) Randomly sample 5 students without replacement from class of 15 male and 15 female students.
g) Randomly select 6 M&M’s from a fun pack without replacement.
h) You attempt free throws with 25% accuracy until the first one is made.
i) Randomly select students until you find one that has not copied homework in school
2. A basketball player has made 80% of his foul shots during the season. Assuming the shots are independent, find the probability that in
tonight’s game he
a) Misses for the first time on his fifth attempt.
b) Makes his first basket on his fourth shot.
c) Makes his first basket on one of his first 3 shots.
d) What is the expected number of shots until he misses?
3. Only 4% of people have Type AB blood.
a) On average, how many donors must be checked to find someone with Type AB blood?
b) What’s the probability that there is a Type AB donor among the first 5 people checked?
c) What’s the probability that the first Type AB donor will be found among the first 6 people?
d) What’s the probability that we won’t find a Type AB donor before the 10 th person?
4. Assume that 13% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below.
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) With what standard deviation?
i) If we keep picking people until we find a lefty, how long do you expect it will take?
5. Suppose a computer chip manufacturer rejects 2% of the chips produced because they fail presale testing.
a) What’s the probability that the fifth chip you test is the first bad one you find?
b) What’s the probability you find at least one bad one within the first 10 you examine?