Transcript AP statistics Chapter 8 Powerpoint notes

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 8:
The Binomial and
Geometric Distributions
Copyright © 2008 by W. H. Freeman & Company
Ex. The number of 3’s rolled by a single die in ten
trials is a binomial random variable
n=10 p=1/6
symbol
X -> B(10,1/6)
Ex. The number of blue tiles chosen from a box
containing 8 blue tiles 2 red tiles and 10 yellow tiles
in 5 trials.
with replacement - n=5, p=0.4
Symbol X-> B(5,0.4)
without replacement – Not binomial; p
changes and observations are not independent
TI-83,84 calculator
• There are two calculator functions that are of great
help in calculating binomial probabilities.
• binompdf(n,p,x) - gives the probability of each
value of x that you input.
• binomcdf(n,p,x) - calculates the cumulative
probability for:
P(x< # of successes) = P(0) + P(1) …
…+ P(# of successes)
Ex.
What is the probability that a couple planning to have 8
children will have 1)exactly 5 boys. 2) at most 5 boys
1) P=0.5, n=8, X=5
symbol X-> B(8,0.5)
Calculator –
P(x=5) = binompdf(8,0.5,5) = 0.21875
2) P=0.5, n=8, X=5 symbol X-> B(8,0.5)
Calculator –
P(x<5) = binomcdf(8,0.5,5) = 0.8555
Binomial Formula
Calculate the probability of 3 sixes in 10 rolls of
a die.
P(success) = 1/6; P(failure) = 5/6
-One possibility; but there are many ways to get 3 sixes
out of ten rolls.
Obs.
outcome
1
6
probability 1/6
2
3
4
5
No
six
ns
ns
6
5/6
5/6
5/6
1/6
6
7
8
9
6
ns
ns
ns
ns
1/6
5/6
5/6
5/6
5/6
The probability for each of the many ways is
equal to (1/6)3 x (5/6)7
10
The binomial coefficient gives the number
of ways this can happen;
= nCr on
calculator
= nCr on
calculator
Ex. What is the probability of having 5 boys out of 7
children?
P(x=5) = 7C5(0.5)5 (0.5)2 = 0.164
TI-83,84 does it for you – binompdf(7,0.5,5)=0.164
Ex. What is the probability that you roll your first 5 on your fifth roll of a die?
“Success” -> roll a 5
P(success) = 1/6
“Failure” -> roll anything else
Roll
1st
Probability 5/6
P(failure) = 5/6
2nd
3rd
4th
5th
5/6
5/6
5/6
1/6
P(x=5) = (5/6)4 x(1/6)1
= 0.08
What is the probability that it takes at most 5 rolls of the die to get a 5?
P(x=1) + P(x=2) + P(x=3) + P(x=4) + P(x=5)
(1/6) + (5/6 x 1/6) +( (5/6)2 x 1/6) + ((5/6)3 x 1/6) +( (5/6)4 x 1/6) =
0.5977
TI-83,84
geometpdf(p,x) and geometcdf(p,x)
P(x=n)
P(x<n)