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Page 1: 4.2B – Finding  B inomial Probabilities

4.2B – Finding Binomial Probabilities

• Binomial Probability FormulanCxpᵡqᶮ ᵡ n MATH PRB nCr x (p)˄x (q)˄(n-x)n=# trials, x=#successes, p=probability of success in 1 trialq= probability of failure in 1 trial

• Calculator2nd VARS 0:binompdf(n,p,x)

Page 2: 4.2B – Finding  B inomial Probabilities

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

Page 3: 4.2B – Finding  B inomial Probabilities

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?

Page 4: 4.2B – Finding  B inomial Probabilities

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?

Page 5: 4.2B – Finding  B inomial Probabilities

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?

Page 6: 4.2B – Finding  B inomial Probabilities

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?

Page 7: 4.2B – Finding  B inomial Probabilities

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?

Page 8: 4.2B – Finding  B inomial Probabilities

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?

Page 9: 4.2B – Finding  B inomial Probabilities

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?

Page 10: 4.2B – Finding  B inomial Probabilities

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

Page 11: 4.2B – Finding  B inomial Probabilities

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)

Page 12: 4.2B – Finding  B inomial Probabilities

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

Page 13: 4.2B – Finding  B inomial Probabilities

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.

Page 14: 4.2B – Finding  B inomial Probabilities

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.

Page 15: 4.2B – Finding  B inomial Probabilities

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.

Page 16: 4.2B – Finding  B inomial Probabilities

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.

Page 17: 4.2B – Finding  B inomial Probabilities

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.

Page 18: 4.2B – Finding  B inomial Probabilities

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.

Page 19: 4.2B – Finding  B inomial Probabilities

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.

Page 20: 4.2B – Finding  B inomial Probabilities

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) =.163P(4) = • 3) Fewer than 2 respond yes.

Page 21: 4.2B – Finding  B inomial Probabilities

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) =.163P(4) = binompdf(4,.41,4) =• 3) Fewer than 2 respond yes.

Page 22: 4.2B – Finding  B inomial Probabilities

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) =.163P(4) = binompdf(4,.41,4) = .028• 3) Fewer than 2 respond yes.

Page 23: 4.2B – Finding  B inomial Probabilities

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) =.163P(4) = binompdf(4,.41,4) = .028 .351 + .163 + .028 =• 3) Fewer than 2 respond yes.

Page 24: 4.2B – Finding  B inomial Probabilities

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) =.163P(4) = binompdf(4,.41,4) = .028 .351 + .163 + .028 = .542• 3) Fewer than 2 respond yes.

Page 25: 4.2B – Finding  B inomial Probabilities

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) =.163P(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) =

Page 26: 4.2B – Finding  B inomial Probabilities

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) =.163P(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

Page 27: 4.2B – Finding  B inomial Probabilities

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) =.163P(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)

Page 28: 4.2B – Finding  B inomial Probabilities

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) =.163P(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

Page 29: 4.2B – Finding  B inomial Probabilities

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) =.163P(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 =

Page 30: 4.2B – Finding  B inomial Probabilities

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) =.163P(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