Definitions• Cumulative – the total from the beginning

to some specified ending point.• Probability Distribution Function (PDF)

– the command on your calculator that will calculate the probability of getting a specific value. i.e. P(x = 6) or P(x = 12)

• Cumulative Distribution Function (CDF) – the command on your calculator that will calculate the probability of getting a specific value or lower. i.e. P(x ≤ 7)

Bernoulli Trials

• Only 2 possible outcomes (success or failure)

• The probability of success is constant.• Each trial is independent.

p = probability of success

q = probability of failure = 1 - p

Which are Bernoulli Trials?a) We roll 50 dice to find the distribution of the

number of spots on the faces.

b) How likely is it that a group of 120 has a majority of people with Type A blood?

c) Deal 5 cards and determine if all are hearts

d) Poll 500 people and determine how many will vote yes on Proposition 5.

e) A company realizes that 10% of its packages are not sealed properly. They examine a case of 24 and determine how likely that more than 3 are unsealed.

Geometric Model

• Must be a Bernoulli trial• Concerned with getting the 1st success• k is the number of trials for 1st success

P (x = k) = q(k – 1)●p

= E(X) = 1/p expected number of trials to get 1st success

PDF vs. CDFPDF = probability distribution function

= the probability of one outcome happening

CDF = cumulative distribution function (add up the probabilities from k down)

X 0 1 2 3

P(x) .13 .24 .55 .08

Example 1: Suppose 75% of drivers wear their seatbelts. We are concerned with how many cars we will have to observe in order to get a driver without a seatbelt on.

a) Is this a Bernouilli trial?

b) P(x = 5)

c) P(x = 4)

d) P(x ≤ 3)

e) What is the expected number of cars that are needed to be observed in order to find a driver without a seatbelt on?

Now we will use our calculator to do these.

Find the distribution menu: 2nd Vars

Scroll down, find geometpdf and/or geometcdf

The function needs the probability of success (P) and k (the number tries to get 1st success)

Type in the command like: geometpdf( p, k )Example: geometpdf(.75, 3) will give you the

probability of it taking 3 tries to get first success if 75% chance of success.

Therefore, geometcdf(.75,4) will give you the probability of getting your 1st success in 4 or less tries.

Binomial Model

• Must be a Bernoulli Trial• Concerned with the probability of getting a

certain number of successes (k) out of n trials. (i.e. what is the probability of getting 3 correct out of 7 questions?)

n = number of trialsk = number of successesp = probability of successq = probability of failure

Binomial Model

!)!(

!

)( )(

kkn

nnCk

k

nwhere

qpk

nkxp knk

Expected Value = = np

Standard deviation = = npq

2) What is the probability of finding 8 drivers wearing seatbelts out of 10 cars? Show all work!

3) What is the mean and standard deviation of the number of drivers wearing seatbelts out of 10? Show all work!

4) What is probability of finding 8 or less drivers wearing seatbelts out of 10?

More Calculator stuff

Binomial pdf and cdf are also in the Distr menu.

Binomial needs 3 statistics, n = total number of trials, p = probability and k = number of successes

For example, the command looks like:binomcdf( n, p, k)

So binomcdf(10, .7, 3) will find the probability of getting 3 or less successes out of 10 tries if .7 is the probability of success.

Example 5: Mr. Kirk is a 78% free throw shooter. Assume each free throw is independent. Find the probability of each. Show the formula and work for questions 1 through 3 and check your answer with your calculator.

1) 1st made free throw is the 3rd free throw attempted.

2) I make 9 out of 123) The mean and standard deviation for 12 free

throws.4) 1st made free throw happens by the 4th free

throw.5) I make at least 6 out of 8 free throws.6) When I expect to make the 1st free throw.

Warm-UpA game consists of spinning a spinner. You have a

23% chance of winning, otherwise you lose. Find the following probabilities:

a) First win on your 3rd try.b) You win 5 times out of 9 plays.c) You win at least 8 times out of 12 plays.d) What is the expected number of plays before

you get your first win?e) You win the first time by the 4th play.f) The expected number of wins and standard

deviation of 12 plays.

ExampleGeorge, Fred, and Tito are professional dart

players. In a certain game, George expects to score 145 with a standard deviation of 11, Fred expects to score 135 with a standard deviation of 8, and Tito expects to score 165 with a standard deviation of 13. Assume each persons score is independent of each other.

a. If George, Fred, and Tito play as a team, what will their expected score and standard deviation be?

b. If George, Fred and Tito’s team score is normally distributed, what is the probability that they score over 470 points?

More Fun with Binomial and Geometric Distributions!

Example #6Suppose a certain type of cereal has a

Derek Jeter card in 20% of the boxes. Find the probability:

a) that the 1st box you open with a Derek Jeter card is the 4th box you open.

b) if you open 6 boxes, you get 2 Derek Jeter cards.

c) if you open 6 boxes, you get at least 2 Derek Jeter cards.

d) it takes you more than 2 boxes to get your first Derek Jeter card.

Example #6 continued

e) What is the mean and standard deviation for the number of Derek Jeter cards if you open 6 boxes?

f) What is the mean number of boxes you will need to open in order to get your first Derek Jeter card?

Binomial Distributions and large number of trials

If n (# of trials) is large, the binomial model starts looking very much like a normal curve.

Conditions to use normal calculations for a binomial model are:

np ≥ 10nq ≥ 10

If the conditions are met, you can use normal calculations to approximate the probability as opposed to BinomialCDF. You can only use the normal approximation if you are finding the probability of a greater than or less than a certain value (in other words, a situation you would use the CDF function)!

Example 6: The Pew Research Center reports that they are actually able to contact only 76% of the randomly selected households drawn for a telephone survey.a) If 800 households are randomly selected, what is the probability that Pew Research Center would actually contact at least 650 of them?

b) If 20 households are randomly selected, what is the probability that Pew would contact 18 of them.

c) What is the probability that the 1st house contacted was on the 5th try?