1. 6: Binomial Probability Distributions
Chapter 6
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6: Binomial Probability Distributions
6.1 Binomial Random Variables
6.2 Calculating Binomial Probabilities
6.3 Cumulative Probabilities
6.4 Probability Calculators
6.5 Expected Value and Variance of Binomial Random Variables
6.6 Using the Binomial Distribution to Help Make Judgments
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Basic Biostat

2. 6.1 Binomial Random Variables
Most popular type of discrete random variable
Binomial random variables are based on a series on independent Bernoulli trials
Bernoulli trial  a random event characterized by “success” or “failure”
Examples
Coin flip (heads or tails)
Survival following cancer (yes or no)
Treatment outcome (success or failure)
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3. Binomial random variables (cont.)
The binomial distributions are a family of distributions identified by these parameters
n  number of trials
p  probability of success for each trial
Let X represent the random number of successes in n independent Bernoulli trials
Notation: X~b(n,p) means “X is distributed as a binomial with parameters n and p”
“Four Patients” Illustrative Example: A treatment is successful 75% of time (p = 0.75). We use it in 4 patients (n = 4). The random number of success in each set of four, X, varies according to this binomial distribution: X~b(4, 0.75)
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4. 6.2 Calculating Binomial Probabilities
The probability X equals x is …
Where
nCx = the binomial coefficient (next slide)
p = probability of success for each trial
q = probability of failure = 1 – p
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5. Binomial Coefficient (“Choose Function”)
where
!  the factorial function: x! = x  (x – 1)  (x – 2)  …  1
Example: 4! = 4  3  2  1 = 24 By definition 1! = 1 and 0! = 1
nCx  the number of ways to choose i items out of n
Example: “4 choose 2”:
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6. 6: Binomial Distributions
Binomial Calculation
Recall the four patients random variable X~b(4, 0.75) Note: q = 1 − 0.75 = 0.25
What is probability of 0 successes?
Pr(X = 0) =nCx px qn–x = 4C0 · · 0.254–0 = 1 · · =
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7. 6: Binomial Distributions
X~b(4,0.75), continued
Pr(X = 1) = 4C1 · · –1
= 4 · · =
Pr(X = 2) = 4C2 · · –2
= 6 · · =
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8. 6: Binomial Distributions
X~b(4, 0.75) continued
Pr(X = 3) = 4C3 · · –3
= 4 · · 0.25 =
Pr(X = 4) = 4C4 · · –4
= 1 · · 1 =
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9. 6: Binomial Distributions
pmf for X~b(4, 0.75)
Tabular form
Graphical form
Successes
Probability
0.0039 1
0.0469 2
0.2109 3
0.4210 4
0.3164
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10. Area Under The Curve (AUC)
The area under the curve represents probability
Pr(X = 2) = 1 × .2109
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11. 6.3 Cumulative Probabilities
Cumulative probability = the probability of that value or less (similar in concept to cumulative frequency as studied in Chapter 3)
Denoted Pr(X  x)
Illustrative example: Cumulative probability function (cdf) for X~b(4, 0.75)
Pr(X  0) = Pr(X = 0) = .0039
Pr(X  1) = Pr(X  0) + Pr(X = 1) = =
Pr(X  2) = Pr(X  1) + Pr(X = 2) = =
Pr(X  3) = Pr(X  2) + Pr(X = 3) = =
Pr(X  4) = Pr(X  3) + Pr(X = 4) = =
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12. Cumulative Probability
Cumulative probability = AUC in left tail
Area under shaded bars in left tail represents: Pr(X  2) =
Bring it on!
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13. 6.5 Expected Value and Variance
These are shortcut formulas apply only to binomial random variables
Four patients illustrative example X~b(4,0.75) μ = np = 4∙0.75 = 3 σ2 = npq = 4∙0.75∙0.25 = 0.75 σ = √σ2 = √0.75 = 0.866
Mean (expected value)
Variance
Standard deviation
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14. 6.6 Using the Binomial to help with Judgments
For the four patients example, we expect 3.
Suppose we observe 2 successes
Does this cast doubt on p = 0.75?
ANS: No, because Pr(X  2) = is not small
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15. 6: Binomial Distributions
Using R
dbinom(x,n,p) = P(X = x) , 0<= x <= n
pbinom(x,n,p) = P(X <= x) 0<= x <= n
1-pbinom(x,n,p) = P(X > x) 0<= x <= n
rbinom(N,n,p) generates N random binomial observations with parameters n and p.
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