4.2B – Finding B inomial Probabilities

1. 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)

2. 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

3. 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?

4. 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?

5. 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?

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? • 3. 75% of small businesses have website. If 10 are selected what is probability exactly 4 have a web site?

7. 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?

8. 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?

9. 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?

10. 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

11. 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)

12. 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

13. 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.

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. 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.

25. 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) =

26. 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

27. 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)

28. 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

29. 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 =

30. 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