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6.3 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. Roll a fair die 100 times. Sometime during the 100 rolls, one corner of the die chips
off. Let W= the number of 5s you roll.
Other examples on pages 388-389
Example page 390-391: 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.
How to find Binomial/Geometric Probabilities
Step 1: State the distribution and the values of interest
Step 2: Perform calculations – show your work!!!
Step 3: Answer the question.
Example Free Lunch pg 396
Local fast-food restaurant is running a “Draw a three, get it free” lunch promotion. After
each customer orders, a touch-screen display shows the message “Press here to win a free
lunch.” A computer program them simulates one card being drawn from a standard deck.
If the chosen card is a 3, the customer’s order is free. Otherwise, the customer must pay
the bill.
a) All 12 players on a school’s basketball team place individual orders at the restaurant.
What is the probability that exactly 2 of them win a free lunch?
b) If 250 customers place lunch orders on the first day of the promotion, what’s the
probability that fewer than 10 win a free lunch?
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.
3. In the board game Monopoly, one way to get out of jail is to roll doubles. Suppose
that this was the only way a player could get out of jail. The random variable of
interest in this example is Y = number of attempts it takes to roll doubles one time.
Each attempt is one trial of the chance process.
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 of Geometric Random Variable
𝒆𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒗𝒂𝒍𝒖𝒆 = 𝝁𝑿 =
𝟏
𝒑
Example Monopoly cont.
In the previous Monopoly example, Y = number of attempts it takes to roll doubles one
time.
(a) Find the probability that it takes 3 turns to roll doubles.
(b) Find the probability that it takes more than 3 turns to roll doubles, and interpret this
value in context.
10% condition
Normal Approximation to Binomial Distributions
*As the number of trials, n, gets larger then the binomial distribution will get closer
to a Normal distribution
𝑵 (𝒏𝒑, √𝒏𝒑(𝟏 − 𝒑))
*Rule of thumb…. use Normal distribution when n satisfies:
𝑛𝑝 ≥ 10 𝐚𝐧𝐝 𝑛(1 − 𝑝) ≥ 10
Example Teens and Debit cards:
In a survey of 506 teenagers aged 14 to 18, subjects were asked a variety of questions
about personal finance. One question asked teens if they had a debit card.
Suppose that exactly 10% of teens aged 14 to 18 have debit cards. Let X = the number of
teens in a random sample of size 506 who have a debit card.
(a) Show that the distribution of X is approximately binomial.
(b) Check the conditions for using a Normal approximation in this setting.
(c) Use a Normal distribution to estimate the probability that 40 or fewer teens in the
sample have debit cards.
Using technology for binomial distribution:
pdf: probability distribution function – assigns a probability to each value of X.
Ti-83/84
2nd DISTR then pick binompdf
Format for parenthesis is 𝑏𝑖𝑛𝑜𝑚𝑝𝑑𝑓(𝑛, 𝑝, 𝑋)
**This calculates binomial distribution for any 𝑃(𝑋 = 𝑘)
cdf: cumulative distribution function – calculates the sum of the probabilities for 0, 1, 2, ….,
up to the value X.
Ti-83/84
2nd DISTR then pick binomcdf
Format for parenthesis is 𝑏𝑖𝑛𝑜𝑚𝑐𝑑𝑓(𝑛, 𝑝, 𝑋)
**This calculates binomial distribution for any 𝑃(𝑋 ≤ 𝑘)
Using technology for geometric distribution:
Same as binomial except –
𝑔𝑒𝑜𝑚𝑒𝑡𝑝𝑑𝑓 (𝑝, 𝑋) 𝑓𝑜𝑟 𝑃(𝑋 = 𝑛)
𝑔𝑒𝑜𝑚𝑒𝑡𝑐𝑑𝑓 (𝑝, 𝑋) 𝑓𝑜𝑟 𝑃(𝑋 ≤ 𝑛)
The probability that it takes more than 𝑛 trials to see the first success in a geometric
setting is:
𝑷(𝑿 > 𝒏) = (𝟏 − 𝒑)𝒏
This is the same as 1 − 𝑔𝑒𝑜𝑚𝑒𝑡𝑐𝑑𝑓(𝑝, 𝑋)