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Section 4.2: BINOMIAL DISTRIBUTIONS
Objective: To be able to find binomial probabilities.
Criteria for a Binomial Random Variable:
1. Each observation can be classified as a success or failure.
2. n is the number of trials and n is fixed.
3. p is the probability of success and p is fixed.
4. The observations are independent.
Binomial Probability Formula:
Let X be a binomial random variable and k be the number of
successes. If X~B(n,p), then
𝑛 𝑘
𝑃 𝑋=𝑘 =
𝑝 ∙ (1 − 𝑝)(𝑛−𝑘) where
𝑘
𝑛
𝑛!
=𝑛 𝐶𝑘 =
which is called the binomial coefficient.
𝑘! 𝑛−𝑘 !
𝑘
(1 - p) = q
q represents the probability of failure.
𝑛 𝑘 (𝑛−𝑘)
𝑃 𝑋=𝑘 =
𝑝 ∙𝑞
(revised with q)
𝑘
The binomial probability distribution is as follows:
𝑿 = 𝒙𝒊
0
1
2
…
n
P(𝑿 = 𝒙𝒊 )
𝒑𝟎
𝒑𝟏
𝒑𝟐
…
𝒑𝒏
Points regarding the binomial distribution:
• A binomial random variable is always discrete.
• To graph use a probability histogram.
• If p = 0.5 then the probability histogram will be bell-shaped.
• If p < 0.5 then the probability histogram will be skewed right.
• If p > 0.5 then the probability histogram will be skewed left.
• Notation: X~B(n,p)
Ex. A large shipment of toys contains 30% that are red, 40%
that are blue and 30% that are yellow. Let X = the number of
red toys. You select 5 toys at random.
a. Does this example meet the criteria for a binomial setting?
Explain.
b. Find P(X = 0)
c. Find P(X = 1)
d. Find P(X = 2)
Calculator notation:
For P(X = k) --- use binompdf(n,p,X)
For P(X ≤ k) --- use binomcdf(n,p,X)
Using your calculator find the following
a. P(X = 3)
b. P(X = 4)
c. P(X = 5)
d. Create the probability distribution of X. Include the
cumulative distribution.
𝑿 = 𝒙𝒊
P(𝑿 = 𝒙𝒊 )
Ex. Let X = the number of successes and that X follows a binomial
distribution with n = 11 and p = 0.72. Answer the following
questions:
a. P(X = 7)
b. P(X < 7)
c. P(X < 9)
d. P(X < 9)
e. P(X < 6)
f. P(X > 8)
g. P(X > 7)
h. P(X > 7)
i. P(X > 10)
Ex. 7 couples buy new homes. It is known that in this large
community 24% of new homes are built with electric heat. Let X
= the number that choose electric heat.
Find the probability that 3 couples choose electric heat. (show
the formula that you would use)
Find the probability that at most 1 couple chooses electric heat.
Find the probability that at least 3 couples choose electric heat.
Find the probability that more than 3 couples choose electric
heat.
Binomial Means, Variance and Standard Deviations
If the random variable X has a binomial distribution with n
observations and the probability of success p, then the mean,
variance and standard deviation of X are
𝜇 = 𝑛𝑝,
𝜎 2 = 𝑛𝑝𝑞
and
𝜎 = 𝑛𝑝𝑞
Using this example again: Couples buy new homes. It is known
that in this large community 24% of new homes are built with
electric heat. Let X = the number that choose electric heat.
What is the expected number of homes that choose electric
heat?
What is the standard deviation of the number of homes that
choose electric heat?