Download AP Statistics – Part IV: Randomness and Probability • Bernoulli trials

AP Statistics – Part IV: Randomness and Probability
Chapter 17: Probability Models (page 386)
• Bernoulli trials – (examples: tossing a coin, looking for defective products,
shooting free throws) Outcomes are either PASS or FAILURE
• Page 387: Calvin and Hobbes
• Geometric Model:
• We are more likely to want to know how long it will take us to achieve a
success. The model that tells us this probability is called the Geometric
Probability Model. Geometric models are completely specified by one
parameter, p, the probability of success, and are denoted Geom(p).
• Geometric Probability Model for Bernoulli trials: Geom(p)
o p = probability of success (and q = 1 – p = probability of failure)
o X = number of trails until the first success occurs
o P(X=x) = qx-1 p
o Expected Value: µ = 1 / p
o Standard deviation: σ = √ (q / p2)
• Trials must be Independent for Bernoulli trials. If you have an infinite supply
of items (ex: the cereal boxes with Tiger Woods cards) then the fact that once
you realize a card is NOT there, the change in probability would be
insignificant. However, if you have a finite population, this does cause the
probabilities to change, making the trials NOT INDEPENDENT.
• The 10% Condition: If we look at less than 10% of the population, we can
pretend that the trials are independent and still calculate probabilities quite
accurately.
o Bernoulli trials must be independent. If that assumption is violated, it
is still OK to proceed as long as the sample is smaller than 10% of the
population.
• Page 389: Step-by-step
• TI TIP: 2nd, Distr, geometpdf( is the “probability density function”. Allows
you to find the probability of an individual outcome. Need p and x = the
number of trials until you get a success.
o geometpdf (p, x)
• TI TIP: 2nd, Distr, geometcdf( is the “cumulative density function”. Finds the
sum of the probabilities of several possible outcomes. Calculates the
probability of finding the first success on or before the xth trial.
o geometcdf (p, x)
• Binomial Model:
• Requires 2 parameters:
o n = number of trials
o p = probability of success.
o Binom (n, p)
• Each different order in which you can have k successes in n trials is called a
combination. Total number of ways this can happen is n or nCk said, “n
choose k”.
k
n!
where n! = n x (n – 1) x … x 1
n = nC k =
k
k! (n – k)!
• The probability of exactly k successes in n trials is n pkqn-k
k
o Expected Value: E(X) = np.
o Standard Deviation: SD(X) = √npq
• A Binomial probability model describes the number of successes in a specified
number of trials.
• Binomial Probability Model for Bernoulli trials: Binom (n, p)
o N = number of trials
o p = probability of success (and q = 1 – p = probability of failure)
o X = umber of successes in n trials.
n!
o P(X = x) = n px qn-x, where n =
x
x
x! (n – x)!
o Mean: µ = np
o Standard Deviation: σ = √npq
• Page 392: Step-by-step
• TI TIP: 2nd, Distr, binompdf( allows you to find the probability of an
individual outcome. You need to define the Binomial model by specifying n
and p, and then indicating the desired number of successes, x.
o Binompdf(n,p,x)
• TI TIP: 2nd, Distr, binomcdf( allows you to find the total probability of getting
x or fewer successes among the n trials.
o Binomcdf(n,p,x)
• TI TIP: To find a “complement”, use 1 – binomcdf(n,p,x)
• Binomial is discrete, giving probabilities for specific counts
• Normal is continuous probability, so you cannot list all possible outcomes for
the probabilities
• Hints:
o Be sure you have Bernoulli trials
o Don’t confuse Geometric and Binomial models
o Don’t use the Normal approximation with small n.