Tommy's Notes Repository

Geometric Distributions

4 min read

How Many Bottle Flips to Go Viral? Y //

Imagine It: The senior talent show is scheduled to take place this afternoon. Michael Senatore is in Stats class practicing his talent - bottle flipping. He tells his friends that the probability of successfully landing the water bottle right side up is 20%. Assume that each water bottle flip is independent.

  1. If he flips the bottle ten times, what is the probability that he makes exactly two flips?

.302 ( binomcpdf(10,.2,2))

You have to know whether or not it is binomial. #1 would get half credit.

BINS is how you check.

  1. Each flip is binary

  2. Each flip is independant

  3. Each flip is n same ?

  4. Each flip is same probability

  5. Whatis the probability that he makes at most two flips? binomcdf(10,.2,2) .67

If you are going to write calculator language you have to do

binomcdf(n=10,p=.2,x=2)

it is also nice to do

P(X⇐2)=binomcdf(n=10,p=.2,x=2)

After the first ten flips, Michael wants to do ten more, but his teacher is getting annoyed with all the noise and tells him he must stop flipping once the bottle lands right side up.

  1. Find the probability that he lands it on the first try.

Geometric distribution - keep going until you stop

.2

  1. Find the probability that he fails on the first try and lands it on the second try.

.4

All Geometric distributions are Skewed right distribution

  1. Find the probability that he lands it on the third try. .008

Let X = the number of flips it takes for Michael to land the bottle right side it up. Fill in the table below.

X123456
P(X).2.16.128.1024.082.066

  1. Find the probability that he lands it on the tenth try. Write a generic rule for finding the probability that he first lands the bottle on the kth flip. p(1-p)^(k-1) .2(1-.2)^(10-1) .2(.8)^9 .8^9 * .2 .028

  2. How many flips do you expect it to take for the bottle to land right side up? Why? something like 5 ? because .2 goes into 1 whole 5 times.

Expected value = mean

If you flip many many many times you should get an expected value of 5 flips before you get a success.

Mx = 1/p = 5

STATS MEDIC

Important Ideas: Geometric Distributions U4, D8

Check Your Understanding!

In a survey of 500 U.S. teenagers aged 14 to 18, subjects were asked a variety of questions about personal finance. One question asked whether teens had a debit card. Suppose that exactly 12% of teens aged 14 to 18 have debit cards. Let X = the number of teens in a random sample of size 500 who have a debit card. a. Explain why X can be modeled by a binomial distribution even though the sample was selected without replacement.

X is binary - they either have a card or they dont have a card X is

b. Use a binomial distribution to estimate the probability that 50 or fewer teens in the sample have debit cards.

c. Justify why X can be approximated by a Normal distribution.

d. Use a Normal distribution to estimate the probability that 50 or fewer teens in the sample have debit cards.

  1. Mason never has a pencil when test day rolls around, Because classmates are tired of having to supply pencils for Mason, only 15% of students will give them a pencil when asked. Today is test day and Mason begins asking randomly selected students for a pencil. Let Y = the number of students Mason asks until they find someone who will give them a pencil.

a. Describe this probability distribution. Be sure to check the appropriate conditions. Binary - give a pencil or dont give a pencil Independant - last person to give a pencil does not affect next person Try until succeed - he just goes until he gets a pencil Same probability throughout - p = 0.15

Geometric

b. What is the probability that the third person asked is the first person who gives Mason a pencil?

P(Y=3)= (.85)^2 * (.15)=.108

c. What is the probability that Mason gets a pencil by the third person they ask? geometcdf(.p = 15, x = 3) P(Y⇐3) = P(y=1) + p(y=2) + P(y=3) = .15+.1275+0.108 = .3855

d. How many people should Mason expect to ask before getting a pencil?

e. Should Mason be surprised if they didn’t receive a pencil until they asked at least 10 people? Calculate a probability to justify your answer,

f. Find the standard deviation for the distribution of the number of people Mason asks for a pencil before finding someone who will give them one.