Transcript Binomial Distribution

AP Statistics
Chapter 8
8.1 – Binomial Distributions
8.2 - Geometric Distributions
The Binomial Setting
• There are 16 games a week in an NFL season
(if no teams are on a bye week). They flip a coin
at each game to decide who gets the ball first.
How many “Tails” will you expect? Will this
always happen?
• If you guess on every question of a 10-question
multiple choice quiz, how well do you think you
will do?
• The previous questions dealt with examples of
random occurrences that take place in a
binomial setting.
Binomial Setting
• 1. Each observation falls into one of just
two categories (often called “success” and
“failure”).
• 2. There is a fixed number, n, of
observations.
• 3. The n observations are all independent.
• 4. The probability of “success”, usually
called p, is the same for each observation.
Binomial Distribution
• The distribution of the count, X, of
successes in the binomial setting…
• B(n, p)
– n # of observations
– p probability of success on any one
observation.
Example
• In 20 rolls of a die, what is the probability of
getting exactly 3 fours?
– Why is this problem difficult to answer based on what
you have already learned?
– Is this a binomial setting?
– You can’t simply use the multiplication rule, because
the fours could be rolled in any 3 of the 20 rolls.
Binomial Coefficient
• The number of ways of arranging k successes
among n observations can be calculated by…
• Read as “n choose k”
• In your calculator, n choose k can be found by
using the command nCr
Finding Binomial Probabilities
• X  binomial distribution
• n  # of observations
• p  prob of success on each observation
Binomial probabilities on the calculator
• P(X = k) = binompdf (n, p, k)
– pdf  probability distribution function 
– Assigns a probability to each value of a
discrete random variable, X.
• P(X < k) = binomcdf (n, p, k)
– cdf  cumulative distribution function 
– for R.V. X, the cdf calculates the sum of the
probabilities for 0, 1, 2 … up to k.
Mean and Standard Deviation
• For a binomial random
variable:
• When n is large, a
binomial distribution can
be approximated by a
Normal distribution.
• We can use a Normal
distribution when.
– np > 10 and n(1 – p) >
10
• If these conditions are
satisfied, then a
binomial distribution can
be approximated by…
The Geometric Setting
• 1. Each observation falls into one of two
categories (“success or “failure”)
• 2. The observations are independent.
• 3. The probability of success, p, is the same for all
observations.
• 4. The variable of interest is the number of trials
required to obtain the first success.
– We do not have a fixed number of trials
– Therefore, the possible values of a geometric random
variable are 1, 2, 3, …
• It is theoretically an infinite set because we may never
observe a “success”
Calculating Geometric Probabilities
• If X has a geometric distribution with probability
p of success and (1 – p) of failure on each
observation, the possible values of X are
1, 2, 3, …
• If n is any one of these values, the probability
that the first success occurs on the nth trial is:
Calculating Geometric Probabilities
• The probability that it takes more than n
trials to the first success is…
Mean and Standard Deviation
• If X is a geometric random variable with
probability of success p on each trial, then the
mean (expected value) of the random variable
is:
• The standard deviation is:
Calculator Functions for Ch 8
• Binomial
– P(X = k)  binompdf(n, p, k)
– P(X < k)  binomcdf(n, p, k)
• Geometric
– P(X < n)  geometcdf(p, n)
• Normal
– P(min< X< max) = normalcdf(min, max, μ, σ)