Chapter 8 Binomial and Geometric Distributions
A binomial setting (or probability distribution)occurs when:
1. Binary? Each observation falls into one of just two categories –
call them “success” or “failure.”
2. Independent? The observations must be independent – result of
one does not affect another.
3. Number? The procedure has a fixed number of trials – we call this
value 𝒏.
4. Success? The probability of success – call it 𝒑 - remains the same
for each observation.
Check the BINS for a binomial probability distribution.
Examples
For each of the following situations, determine whether the given
random variable has a binomial distribution. Justify your answer.
1. Shuffle a deck of cards. Turn over the top card. Put the card
back in the deck, and shuffle again. Repeat this process 10 times.
Let X=the number of aces you observe.
2. Choose three students at random from your class. Let Y=the
number who are over 6 feet tall.
3. An engineer chooses an SRS of 10 switches from a shipment of
10,000 switches. Suppose that (unknown to the engineer) 10%
of the switches in the shipment are bad. The engineer counts the
number X of bad switches in the sample.
*Look closely at independence.
Other examples on pages 514-515
Example 8.5 page 517: Read in pairs. How would you find 𝑷(𝑿 = 𝟑)?
Notation for binomial probability distribution
𝑛 denotes the number of fixed trials
𝑘 denotes the number of successes in the 𝑛 trials
𝑝 denotes the probability of success
1 – 𝑝 denotes the probability of failure
𝐵(𝑛, 𝑝) denotes a binomial probability distribution
Binomial Coefficient: number of ways of arranging 𝑘 successes
𝑛!
𝑛
( )=
𝑘
𝑘! (𝑛 − 𝑘)!
Binomial Probability:
𝑛
𝑃(𝑋 = 𝑘) = ( ) 𝑝𝑘 (1 − 𝑝)𝑛−𝑘
𝑘
Mean and Standard Deviation of Binomial Random Variable
𝝁𝑿 = 𝒏𝒑
𝝈𝑿 = √𝒏𝒑(𝟏 − 𝒑)
These are only for binomial distributions, not other discrete random variables.
Example 6 pg 519
The number X of switches that fail inspection in Example 3 has
approximately the binomial distribution with n=10 and p=0.1. What
is the probability that no more than one switch fails?
A geometric setting (or probability distribution) occurs when:
1. Binary? Each observation falls into one of just two categories –
call them “success” or “failure.”
2. Independent? The observations must be independent – result of
one does not affect another.
3. Trials? The variable of interest is the number of trials required to
obtain the first success.
4. Success? The probability of success – call it 𝒑 - remains the same
for each observation.
*the geometric is also called a “waiting-time” distribution
Check the BITS for a geometric probability distribution
Examples:
1. A game consists of rolling a single die. The event of interest is
rolling a 3; this event is called a success. The random variable is
defined as X=the number to trials until a 3 occurs.
2. Suppose you repeatedly draw cards without replacement from a
deck of 52 cards until you draw and ace.
*pg. 541
Example 8.15 page 540-541
Calculate probabilities for X=1, X=2, and X=3, for the rolling a single
die example above.
Notation for geometric probability distribution
𝑛 denotes the number of trials required to obtain the first success
𝑝 denotes the probability of success
1 – 𝑝 denotes the probability of failure
Geometric Probability:
𝑷(𝑿 = 𝒏) = (𝟏 − 𝒑)𝒏−𝟏 𝒑
Mean and Standard Deviation of Geometric Random Variable
𝟏
𝝁𝑿 =
𝒑
𝝈𝑿 = √
(𝟏 − 𝒑)
𝒑𝟐
Example 8.18
Glenn likes the game at the state fair where you toss a coin into a saucer.
You win if the coin comes to rest in the saucer without sliding off. Glenn has
played this game many times and has determined that on average he wins 1
out of every 12 times he plays. He believes that his chances of winning are the
same for each toss. He has no reason to think that his tosses are not
independent. Let X be the number of tosses until a win. Determine whether
this is Binomial or Geometric, then find the mean and standard deviation.