4.2B – Finding Binomial Probabilities
• Binomial Probability Formula
nCxpᵡqᶮ ᵡ
n MATH PRB nCr x (p)˄x (q)˄(n-x)
n=# trials, x=#successes,
p=probability of success in 1 trial
q= probability of failure in 1 trial
• Calculator
2nd VARS 0:binompdf(n,p,x)
Probability Distribution
• List of possible values of x with each
corresponding probability in a table.
• Example: 7 participants (#trials) 36% answered
yes (p=.36) (q=.64) Binomial probability
distribution for # responding yes.
P(0) = ₇C₀(.36)⁰(.64)⁷≈.044 = binompdf(7,.36,0)
P(1) = ₇C₁(.36)(.64)⁶≈.173 = binompdf(7,.36,1)
P(2) = ₇C₂(.36)²(.64)⁵≈.292 = binompdf(7,.36,2) etc.
x
0
1 2
3
4
5
6
7
p(x) .044 .173 .292 .274 .154 .052 .010 .001
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ =
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65)
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
₂₅₀C₁₇₈(.71)˄178(.29)˄72 =
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
₂₅₀C₁₇₈(.71)˄178(.29)˄72 = binompdf(250,.71,178)
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households.
What is probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog.
Select 250 people. What is probability exactly
178 use more than 1 topping?
₂₅₀C₁₇₈(.71)˄178(.29)˄72 =
binompdf(250,.71,178)≈.056
• 3. 75% of small businesses have website. If 10
are selected what is probability exactly 4 have a
web site?
Examples:
• 1. 58% own gas grills. Select 100 households. What is
probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog. Select 250
people. What is probability exactly 178 use more than
1 topping?
₂₅₀C₁₇₈(.71)˄178(.29)˄72 = binompdf(250,.71,178)≈.056
• 3. 75% of small businesses have website. If 10 are
selected what is probability exactly 4 have a web site?
₁₀C₄(.75)⁴(.25)⁶
Examples:
• 1. 58% own gas grills. Select 100 households. What is
probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog. Select 250
people. What is probability exactly 178 use more than
1 topping?
₂₅₀C₁₇₈(.71)˄178(.29)˄72 = binompdf(250,.71,178)≈.056
• 3. 75% of small businesses have website. If 10 are
selected what is probability exactly 4 have a web site?
₁₀C₄(.75)⁴(.25)⁶ = binompdf(10,.75,4)
Examples:
• 1. 58% own gas grills. Select 100 households. What is
probability exactly 65 will own one?
₁₀₀C₆₅(.58)⁶⁵(.42)ᶟ⁵ = binompdf(100,.58,65) ≈.03
• 2. 71% use more than 1 topping on hot dog. Select 250
people. What is probability exactly 178 use more than
1 topping?
₂₅₀C₁₇₈(.71)˄178(.29)˄72 = binompdf(250,.71,178)≈.056
• 3. 75% of small businesses have website. If 10 are
selected what is probability exactly 4 have a web site?
₁₀C₄(.75)⁴(.25)⁶ = binompdf(10,.75,4) ≈ .016
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2)
• 2) at least 2 respond yes
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²
• 2) at least 2 respond yes
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) =
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) =
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) =
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 =
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability
that they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes.
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability that
they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes. = P(0) + P(1)
P(0)=binompdf(4,.41,0) =
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability that
they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes. = P(0) + P(1)
P(0)=binompdf(4,.41,0) = .121
Examples
• 41% of women choose reading as favorite leisure
activity. 4 are selected What is the probability that
they answer yes it is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes. = P(0) + P(1)
P(0)=binompdf(4,.41,0) = .121 P(1) = binompdf(4,.41,1)
Examples
• 41% of women choose reading as favorite leisure activity. 4
are selected What is the probability that they answer yes it
is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes. = P(0) + P(1)
P(0)=binompdf(4,.41,0) = .121 P(1) = binompdf(4,.41,1) =.337
Examples
• 41% of women choose reading as favorite leisure activity. 4
are selected What is the probability that they answer yes it
is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes. = P(0) + P(1)
P(0)=binompdf(4,.41,0) = .121 P(1) = binompdf(4,.41,1) =.337
.121 + .337 =
Examples
• 41% of women choose reading as favorite leisure activity. 4
are selected What is the probability that they answer yes it
is their favorite?
• 1) exactly 2 respond yes.
P(2) = binompdf(4,.41,2) = ₄C₂(.41)²(.59)²≈.351
• 2) at least 2 respond yes = P(2) + P(3) + P(4)
P(2)=.351 P(3) = binompdf(4,.41,3) =.163
P(4) = binompdf(4,.41,4) = .028
.351 + .163 + .028 = .542
• 3) Fewer than 2 respond yes. = P(0) + P(1)
P(0)=binompdf(4,.41,0) = .121 P(1) = binompdf(4,.41,1) =.337
.121 + .337 = .458