ap statistics: section 8.1a binomial probability
TRANSCRIPT
AP Statistics: Section 8.1A
Binomial Probability
There are four conditions to a binomial setting:
1. Each observation falls into one of just two categories: _________ or ________.
2. There is a ______ number of observations, __.
3. These n observations are all ____________.
4. The probability of success, __, is _________ for each observation.
failure success
finite n
tindependen
constant p
If data are produced in a binomial setting, then the random variable X = the number of successes is called a binomial random
variable and the probability distribution of X is called a binomial distribution which is
abbreviated ______.),( pnB
Example 1: Determine if each of the following situations is a
binomial setting. If so, state the probability distribution for X. If
not, state which of the 4 conditions above is not met.
Situation 1: Blood type is inherited. If both parents carry genes for the O and A blood types, each child has probability 0.25 of getting two O genes and so having blood type O. A couple’s 5 different children inherit
independently of each other. Let X = number of children with type O blood.
)25,.5(B
Situation 2: Deal 10 cards from a shuffled deck and let X = the number of red cards.
51
25
51
26,
52
26
constant.not are iesprobabilit No,
or
Situation 3: An engineer chooses a SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. Let X = the number of bad switches in the sample.
)1,.10(B
****When choosing an SRS from a population that is much larger than the sample, the observations are considered independent.
Binomial Probability: If X has a binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2, 3, . . . , n. If k is any one of these values, then
knk ppk
nkxP
)1)(()(
The notation is read n choose k and means the number of possible
ways to choose k objects from a group of n objects.
It is also written _____
k
n
knC
:84/83TI
k
n
)!(!
!
knk
n
rCn: 3 PRB MATH
(3)(2)(1)2)-1)(n-n(n means and factorialn read is n!notation The
720123456 6! So
10! definitionBy
! :4 PRB MATH :84/83TI
Example 2 (combinations): How many different ways can we choose a
subcommittee of size 3 from a student council that has 7 members?
356
210
!4!3
!7
3
7
3537 rnC
Example 3: You randomly guess the answers to10 multiple choice questions which have 5 possible answers. What is the probability of
getting exactly 6 correct answers?
6 with x)2,.10( B
)8)(.2(.
6
10 46 0055.
Binomial Probability on the TI83/84:
ENTER fbinomialpd:0 DISTR VARS 2nd
),,( xpnfbinomialpd
Example 4: Consider situation 3 in example 1. Find the probability that in an SRS of size 10, no
more than 1 switch fails.
x where)1,.10( B 1or 0
.7361
3874.)1,1,.10(
3487.)0,1,.10(
fbinomialpd
fbinomialpd
Example 5: Corinne is a basketball player who makes 75% of her free throws over the course of a season. In a big game, Corinne shoots 12 free throws and makes only 5 of them. Is it unusual for Corinne to perform this poorly?Note: We actually want the probability of making a basket on at most 5 free throws.
Cumulative Binomial Probability on the TI83/84:
5or 0,1,2,3,4 x where)75,.12( B
fbinomialcd:A DISTR VARS 2nd
value)largest x ,,( pnfbinomialcd
0143.)5,75,.12( binmialcdf
Any difference in answers between the manual calculation and using your calculator is due to rounding
error.