1. Teacher
Introductory Statistics
Lesson 4.2 B
Objective:
SSBAT find binomial probabilities using the binomial probability formula.
Standard: S2.5B

2. Binomial Probability Formula
Review
Binomial Probability Formula
The probability of exactly x successes in n trials is:
P(x) = nCx px qn-x
Where:
n = Number of times a trial is Repeated
p = Probability of Success P(S)
q = Probability of Failure P(F)
x = Represents a count of the number of successes in n trials: x = 0, 1, 2, 3, …, n

3. Probability of an OR statement  Add the probabilities
Review:
Probability of an OR statement  Add the probabilities
P(Rolling a 2 or 3) = P(2) + P(3)
P(Rolling at least a 4) = P(4) + P(5) + P(6)
P(Rolling a number < 3) = P(1) + P(2)

4. P(at least 2) = P(2) + P(3) + P(4)
Examples:
A survey indicates that 41% of women in the U.S. consider reading their favorite leisure-time activity. You randomly select 4 U.S. women. Find the probability that at least 2 of them said reading is their favorite leisure-time activity.
n = 4, p = .41, q = 1 – 0.41 = 0.59, x = 2, 3, or 4
P(at least 2) = P(2) + P(3) + P(4)
 Use the formula to find P(2), P(3), and P(4)
 Then add the 3 probabilities together
P(x) = nCx px qn-x

5. Teacher
P(x) = nCx px qn-x
#1 Continued.
n = 4, p = .41, q = 1 – 0.41 = 0.59, x = 2, 3, or 4
P(2) + P(3) + P(4)
P(2) = 4C2 (.41)2 (.59)2
P(3) = 4C3 (.41)3 (.59)1
P(4) = 4C4 (.41)4 (.59)0 = = =
P(at least 2) =
P(at least 2) =

6. P(fewer than 2) = P(0) + P(1)
P(x) = nCx px qn-x
A survey indicates that 21% of men in the U.S. consider fishing their favorite leisure-time activity. You randomly select 5 U.S. men. Find the probability that fewer than 2 of them said fishing is their favorite leisure-time activity.
n = 5, p = .21, q = 1 – 0.21 = 0.79, x = 0 or 1
P(fewer than 2) = P(0) + P(1)
 Use the formula to find P(0) and P(1)
 Then add the 2 probabilities together

7. P(0) = 5C0 (.21)0 (.79)5 P(1) = 5C1 (.21)1 (.79)4 = 0.308 = 0.409
Teacher
P(x) = nCx px qn-x
#2 Continued.
n = 5, p = .21, q = 0.79, x = 0 or 1
P(0) + P(1)
P(0) = 5C0 (.21)0 (.79)5
P(1) = 5C1 (.21)1 (.79)4 = =
P(fewer than 2) =
P(fewer than 2) =

8. P(4 or more) = P(4) + P(5) + P(6) + P(7)
P(x) = nCx px qn-x
Micro-fracture knee surgery has a 75% chance of success on patients with degenerative knees. The surgery is performed on 7 patients. Find the probability that 4 or more patients will have success.
n = 7, p = .75, q = 1 – 0.75 = 0.25, x = 4, 5, 6, or 7
P(4 or more) = P(4) + P(5) + P(6) + P(7)

9. Teacher
#3 Continued.
n = 7, p = .75, q = 0.25, x = 4, 5, 6, 7
P(4) + P(5) + P(6) + P(7)
P(4) = 7C4 (.75)4 (.25)3
P(5) = 7C5 (.75)5 (.25)2
P(6) = 7C6 (.75)6 (.25)1
P(7) = 7C7 (.75)7 (.25)0
P(x) = nCx px qn-x = = = =
P(4 or more) =
P(4 or more) =

10. P(at most 2) = P(0) + P(1) + P(2)
P(x) = nCx px qn-x
You are taking a multiple-choice quiz that consists of 5 questions. Each question has 4 possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each questions. Find the probability of guessing at most 2 questions correctly.
n = 5, p = ¼ = 0.25, q = 1 – 0.25 = 0.75, x = 0, 1, 2
P(at most 2) = P(0) + P(1) + P(2)

11. Teacher
#4 Continued.
n = 5, p = .25, q = 0.75, x = 0, 1, 2
P(0) + P(1) + P(2)
P(0) = 5C0 (.25)0 (.75)5
P(1) = 5C1 (.25)1 (.75)4
P(2) = 5C2 (.25)2 (.75)3
P(x) = nCx px qn-x = = =
P(at most 2) =
P(at most 2) =

12. Complete Worksheet 4.2B