Binomial Distributions

1
Binomial Distributions

• Mr. Luitwieler
• AP Statistics

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Introduction

• What IS a Binomial Distribution? How do I know I
  have THIS type of distribution?

• Once I find out WHAT it is, how do I USE this
  type of distribution?

• Mean and Standard Deviation of this distribution?

• Normal Approximation to the binomial
  distribution?

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THE BINOMIAL SETTING

• Only TWO (discrete) possible outcomes denoted
  success and failure

• There is a FIXED number of n events.

• The events are all INDEPENDENT.

cdf
n

• The probability of success, (called p), is the
  SAME for each outcome.
• Note x 0 is a viable RESULT!

p
np
s
nCk
gt10
m

• ALL OF THE ABOVE MUST EXIST FOR IT TO BE A
  BINOMIAL DISTRIBUTION!

pdf
4
Binomial Setting?

• According to the Textbook of Medical Physiology,
  by Arthur Guyton, 9 of the population has blood
  type B. Suppose we choose 18 people at random
  from the population and test the blood type of
  each.
• Is this a Binomial Distribution?

• YES Either the individual has type B or not.
• Each person has a 9 chance of success.
• Independence can be approximated since the
  population is LARGE and the sample size is
  SMALL!n is 10, or less, of N

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The ACTUAL Distribution

• Shape
• p.75
• n12
• Closerp is to .5, what happens to shape?

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Binomial Distributions n 10
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Probability Distribution Function

• binomialpdf(n,p,k) gives you the probability,
  given n possible outcomes and a proportion of
  success (p), of a SPECIFIC outcome (x k).
• P(x10), given 45 chance of success and 15
  outcomes
• binomialpdf(15,.45,10) 0.0515

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Cumulative Distribution Function

• binomialcdf(n,p,k) finds the cumulative
  probability of all outcomes up to and including
  the outcome of x k.
• P(x lt 11), given 15 outcomes, and 45 chance of
  success
• binomialcdf(15,.45,10) 0.9745
• Note this is a DISCRETE distribution

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Binomial Formula

• The FORMULA on formula sheet! ?
• The binomial coefficient

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Mean and Standard Deviation

• Mean of a Binomial Distribution on formula
  sheet! ?
• Standard Deviation
• OR

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Example Problem

• A home alarm system has detectors covering n
  zones of the house. Suppose the probability is
  0.7 that a detector sounds an alarm when an
  intruder passes through its zone, and this
  probability is the same for each detector. The
  alarms operate independently. An intruder enters
  the house and passes through all the zones.
• What is the probability that an alarm sounds if n
  3?
• What is the probability that an alarm sounds if n
  6? Does P(x) double?
• How large must n be to have the porobability of
  an alarm sounding be at least .99?

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Solution

• Part c Algebra!

• Parts a,b
• P(an alarm sounding) 1- P(NO alarm
  soundingP(X0) binomialpdf(3,.7,0)
  .0271-0.27 .973
• Same, except n 6, so P(X0)
  binomialpdf(6,.7,0) 7.29E-41-7.29E-4
  .999271, NO it doesnt doublewhy not?

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Normal Approximation

• When n is LARGE, then a binomial distribution
  can be approximated by a normal distribution the
  mean and s.d. of which is the mean and s.d. of
  the binomial distribution.

• The rule of thumb n is considered large ifnp gt
  10 and n(1-p) gt 10.

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CALCULATOR SKILLS!

• Make sure you look at the Technology Toolbox on
  p.456.
• p.458 BINOMIAL SIMULATIONS