Geometric Distributions

Geometric Distributions

Multiple Choice

Which of the following is true about geometric distributions?

• Geometric distributions are approximately skewed right.
• Geometric distributions are approximately symmetric.
• Geometric distributions are approximately skewed left.
• The shape could be symmetric or skewed.

Multiple Choice

A geometric probability model follows a Bernoulli trial and:

• Has a set sample size n.
• Is looking for first success

Multiple Choice

If we want to calculate the probability of rolling a 6 for the first time on the 6^th roll of a die, would we use the geometric or the binomial distribution?

• geometric

• binomial

Geometric Distribution Probability Density Function (pdf)

• Trials can only have two outcomes (Success or Failure).

• Use Probability Density Function (pdf) when you want to know the probability of your FIRST success being the nth trial, given the probability of a success of each individual trial is p.

• Can be found using P(x=n)=(1-p)^n-1p

• Where p= probability of success and n=number of trials.

Geometric Distribution pdf Example

Find the probability of your first heads being on the 5th coin flip.

p=0.5 (probability of getting heads on a coin flip)

n=5 (FIRST success on the 5th trial)

P(x=n)=(1-p)^n-1p

P(x=5)=(1-0.5)^5-1(0.5)

P(x=n)=(0.5)⁴0.5)= .031

Calculator: 2nd+VARS->geometpdf(0.5,5)

Multiple Choice

What is the probability that a family doesn't have a baby girl until their third child?

• 0.13
• 0.15
• 0.05
• 0.88

Multiple Choice

The probability that Jalen makes a free throw is 75%. What is the probability he makes his first free throw on his third attempt?

• geometcdf(3, 0.75)
• geometpdf(0.75, 3)
• geometpdf(3, 0.75)
• geometcdf(0.75, 3) 

Multiple Choice

The owner of a pet store is trying to decide whether to discontinue selling specialty clothes for pets. She suspects that only 4% of the customers buy specialty clothes for their pets. What is the probability that she does NOT sell a garment until the 7th customer? Assume each customer is independent.

• 0.033
• 0.027
• 0.0313
• Not here

Multiple Choice

The local Chipotle is running a promotion where each large drink cup has a large pull-off tab that reveals a prize (free burrito!) or the message "Try again next time."  Chipotle claims that 25% of the cups have tabs with a  prize.  You visit Chipotle to try to win.  What is the probability that you win on your third visit?

• 0.86
• 0.14
• 0.42
• 0.58

Multiple Choice

20% of C-Hill students have experienced feelings of math anxiety. The counselors are interviewing students. What is the probability that the first student Mrs. Stout counsels for math anxiety is the tenth student she sees? 

• 0.02
• 0.89
• 0.03
• 0.11

Multiple Choice

Only 4% of people have type AB blood.  What is the probability the first person with AB blood at the blood drive will be the 12th donor?

• .0001
• .0255
• .6382
• .2553

Multiple Choice

Given 13% of people are left-handed.  What is the probability the first left-handed person will be the sixth person polled?  

• .0564
• .1116
• .0001
• .0648

Geometric Distribution Continuous Density Function (cdf)

Trials can only have two outcomes (Success or Failure).

Use Probability Density Function (pdf) when you want to know the probability of your FIRST success being ON or BEFORE the nth trial, given the probability of a success of each individual trial is p.

Can be found using P(x<=n)=P(x=1) + P(x=2)+ ...P(x=n)

Where: P(x=n) = (1-p)^n-1p

p= probability of success and n=number of trials.

Geometric cdf Example

Find the probability of your first heads being ON OR BEFORE the 5th coin flip.

p=0.5 (probability of getting heads on a coin flip)

n=5 (FIRST success on or BEFORE the 5th trial)

P(x=n)=(1-p)^n-1p

P(x<=5)= P(x=1)+P(x=2)+P(x=3) +P(x=4) +P(x=5)

p(x<=5)=(1-0.5)⁰(0.5)+(1-0.5)¹(0.5) +(1-0.5)²(0.5)+(1-0.5)³(0.5) + (1-0.5)⁴(0.5)

p(x<=5)=0.5+0.25+0.125+0.625+0.031=0.96

Calculator: 2nd+VARS->geometcdf(0.5,5)=0.96

Multiple Choice

In hockey, 5 players from each team take shots if the game ends in a tie (a shootout). The chances of making a goal in a shootout is low, only 14%. If the Caps and Penguins go into a shootout, what is the probability that the first goal scored is after the first four shots?

• 0.64
• 0.47
• 0.45
• 0.55

Multiple Choice

Find the probability that LeBron James makes a three-point shot on or before his seventh attempt assuming he makes 31.5% of all three-pointers.

• 0.97
• 0.03
• 0.07
• 0.93

Multiple Choice

Peyton Manning completes 67% of his passes. What is the probability that it takes more than four throws to complete his first pass in a game?

• 0.99
• 0.01
• 0.04
• 0.96

Multiple Choice

If Mobius draws 7 cards out of a deck with replacement, what is the probability that he gets the first heart in 6 attempts or fewer?

• 0.18
• 0.82
• 0.06
• 0.76

Multiple Choice

If the probability that a house in your neighborhood has a dog is 60%, what is the chance that you find a house with a dog before the fifth house?

• 0.97
• 0.99
• 0.02
• 0.04

Multiple Choice

Lilah flips a coin. What is the chance the coin comes up with the first tails in fewer than 5 flips?

• geometcdf(1/2, 5)
• geometpdf(1/2, 5)
• geometcdf(1/2, 4)
• geometpdf(1/2, 4)

Multiple Choice

According to the BMV, about 93% of all people who take the driving test pass.  What is the probability it will take someone more than two times to pass the test?

• .9300
• .0651
• .9951
• .0049

Multiple Choice

The probability that a salesman will make a sale on any given call is .23.  Find the probability that his first sale will occur on his fourth or fifth call.

• .1050
• .0809
• .1859
• .8141