*
Probability(C14-C17 BVD)
C17: Binomial and Geometric Probability Models
* Many probability questions on AP exam fit a
binomial model
* BINS
* B – Binary? Outcomes can be defined as
success/fail.
* I – Independent? Knowing result of one trial does
not effect result of another trial
* N – Number? Number of trials n fixed in advance
* S – Success Same? Probability of success (p) is the
same for each trial.
* X = number of successes in n trials
* P(X=x) = nCx (px) (1-p)n-x
* Mean µ = np
* Standard deviation σ = sqrt(npq)
* Notation: Binom(n,p)
* Binompdf(n,p,X)
* Binomcdf(n,p,X) – up to X successes
* Complement rule (1- opposite) very handy
* See page 331-332 for example
*
* Many probability questions on AP exam fit a
geometric model
* BITS
* B – Binary?
Outcomes can be defined as
success/fail.
* I – Independent?
Knowing result of one trial does
not effect result of another trial
* T – Trials? Asking how many trials until success
* S – Success Same? Probability of success (p) is the
same for each trial.
*
* X = number of trials until success
* P(X=x) = (1-p)x-1p
* Mean µ = 1/p
* Standard deviation σ = sqrt(q/p2)
* Notation: Geomet(n,p)
* Geometpdf(p,X)
* Geometcdf(p,X) – 1st success on or before trial x
* Complement rule (1- opposite) very handy
* See page 328 for example
*
* Technically I in BINS or BITS is not met when in a
sampling situation – when you draw from a finite
population, the probability of the next event
changes slightly (think of drawing Ace then Ace in a
deck of cards)
* However, if the sample is smaller than 10% of the
population, the probabilities change slightly enough
that they are almost as if they didn’t change at all.
* So: 1 0 % C o n d i t i o n – if sample is < 10% pop,
we can still use Binomial/Geometric probability
models and procedures
*
*A Binomial model is approximately
Normal if we expect at least 10
successes and 10 failures
*(np and np ≥ 10)
*If that is true, use N(µ,σ) instead of
Binom(n,p,X) with Mean µ = np and
Standard deviation σ = sqrt(npq)
*C6 again!
*