Transcript 4.2 Binomial Distributions - 31-ICT

Binomial Distributions
Binomial Experiments
• Probability experiments for which the
results of each trial can be reduced to
two outcomes: success and failure.
• When a basketball player attempts a
free throw, he or she either makes the
basket or does not.
• Probability experiments such as these
are called binomial experiments.
Binomial Probability Distribution

A Fixed Number of Observations (trials), n


A Binary Outcome




15 tosses of a coin; 20 patients; 1000 people
surveyed
Head or tail in each toss of a coin; disease or no
disease
Generally called “success” and “failure”
Probability of success is p, probability of failure is
1–p
Constant Probability for each observation

Probability of getting a tail is the same each time we
toss the coin
Notation for Binomial Experiments
Symbol
Description
n
The number of times a trial is repeated.
p = P(S)
The probability of success in a single
trial.
q = P(F)
The probability of failure in a single trial
(q = 1 – p)
X
The random variable represents a
count of the number of successes in n
trials: x = 0, 1, 2, 3, . . . n.
Example
• We pick a card from a standard deck of
cards, and note whether it is a club or not,
and replace the card. We repeat the
experiment 5 times.
•
•
•
•
n=5
p = P(S) = ¼
q = P(F) = ¾
Possible values of the random variable are
0, 1, 2, 3, 4, and 5.
Binomial Experiments
• Decide whether the experiment is a
binomial experiment:
• A binomial experiment specify the values
of n (number of times a trial is repeated), p
(Probability of Success), q (Probability of
Failure) and list the possible values of the
random variable, x.
Example
A certain surgical procedure has an
85% chance of success.
A doctor
performs
on
patients.
the
procedure
The
random
eight
variable
represents the number of successful
surgeries.
Binomial Experiment
• Each surgery represents one trial. There
are eight surgeries, and each surgery is
independent of the others.
• Only two possible outcomes for each
surgery - either the surgery is a success or
it is a failure.
n=8
p = 0.85
q = 1 – 0.85 = 0.15
x = 0, 1, 2, 3, 4, 5, 6, 7, 8
Example
A jar contains five red marbles, nine blue
marbles and six green marbles.
Select
randomly three marbles from the jar, without
replacement.
The
random
variable
represents the number of red marbles.
Not A Binomial Experiment
• Each marble selection represents one trial
and selecting a red marble is a success.
• When selecting the first marble, the
probability of success is 5/20. However
because the marble is not replaced, the
probability of further trials is no longer
5/20.
• Trials are not independent.
Binomial Probability Formula
P( x)  Cx p q
n
x
n x
n!
x n x

pq
(n  x)!x!
Example: Binomial Probabilities
• A six sided die is rolled 3 times. Find the
probability of rolling exactly one 6.
Roll 1
Roll 2
Roll 3
# of 6’s
Probability
(1)(1)(1) = 1
3
1/216
(1)(1)(5) = 5
2
5/216
(1)(5)(1) = 5
2
5/216
(1)(5)(5) = 25
1
25/216
(5)(1)(1) = 5
2
5/216
(5)(1)(5) = 25
1
25/216
(5)(5)(1) = 25
1
25/216
(5)(5)(5) = 125
0
125/216
Frequency
Example: Binomial Probabilities
• Three outcomes that have exactly one six
• Each has a probability of 25/216
• Probability of rolling exactly one six is 3(25/216)
≈ 0.347.
• Binomial Probability Formula (n = 3, p = 1/6, q = 5/6
and x = 1). Probability of rolling exactly one 6 is:
P( x)  n C x p q
x
Binomial Probability Formula
n x
n!

p x q n x
(n  x)!x!
Example: Binomial Probabilities
3!
1 1 5 31
P (1) 
( ) ( )
(3  1)!1! 6
6
1 5 2
 3( )( )
6 6
1 25
 3( )(
)
6 36
25
 3(
)
216
25

 0.3 4 7
72
Binomial Probability Distribution
• By listing the possible values of x with
the corresponding probability of each,
we
can
construct
Probability Distribution.
a
Binomial
Constructing a Binomial Distribution
In a survey, a company asked their
workers and retirees to name their expected
sources of retirement income. Seven
workers who participated in the survey were
asked whether they expect to rely on
Pension for retirement income. 36% of the
workers responded that they rely on Pension
only. Create a binomial probability
distribution.
Constructing a Binomial Distribution
P(0)7 C0 (0.36) (0.64)  0.044
x
P(x)
0
0.044
P(1)7 C1 (0.36)1 (0.64)6  0.173
1
0.173
P(2)7 C2 (0.36)2 (0.64)5  0.292
2
0.292
P(3)7 C3 (0.36)3 (0.64)4  0.274
3
0.274
4
0.154
5
0.052
P(5)7 C5 (0.36)5 (0.64)2  0.052
6
0.010
P(6)7 C 6(0.36)6 (0.64)1  0.010
7
0.001
0
7
P(4)7 C4 (0.36) (0.64)  0.154
4
3
P(7)7 C7 (0.36) (0.64)  0.001
7
0
P(x) = 1
Notice all the probabilities are between 0
and 1 and that the sum of the probabilities is
1.
Finding a Binomial Probability Using a Table
• Fifty percent of working adults spend less than 20 minutes
commuting to their jobs. If you randomly select six working
adults, what is the probability that exactly three of them
spend less than 20 minutes commuting to work?
• Using the distribution for n = 6 and p = 0.5, we can find the
probability that x = 3
Population Parameters of a
Binomial Distribution
Mean:  = np
Variance: 2 = npq
Standard Deviation:  = √npq
Example
• In Murree, 57% of the days in a year are cloudy.
Find the mean, variance, and standard deviation
for the number of cloudy days during the month
of June.
Mean:
 = np = 30(0.57) = 17.1
Variance: 2 = npq = 30(0.57)(0.43) = 7.353
Standard Deviation:  = √npq = √7.353 ≈2.71
Problem 1
Four fair coins are tossed simultaneously.
Find the probability function of the random
variable X = Number of Heads and compute
the probabilities of obtaining no heads,
precisely 1 head, at least 1 head, not more
than 3 heads.
Problem 2
If the Probability of hitting a target in
a single shot is 10% and 10 shots are
fired
independently.
What
is
the
probability that the target will be hit at
least once?
Problem 3
If the Probability of hitting a target in
a single shot is 5% and 20 shots are
fired
independently.
What
is
the
probability that the target will be hit at
least once?
Problem 5
Let X be the number of cars per minute
passing a certain point of some road between
8 A.M and 10 A.M on a Sunday. Assume that
X has a Poisson distribution with mean 5.
Find the probability of observing 3 or fewer
cars during any given minute.
Problem 7
In 1910, E. Rutherford and H. Geiger
showed experimentally that number of alpha
particles emitted per second in a radioactive
process is random variable X having a
Poisson distribution. If X has mean 0.5. What
is the probability of observing 2 or more
particles during any given second?
Problem 9
Suppose that in the production of 50л
resistors, non-defective items are those that
have a resistance between 45л and 55л
and
the probability of being defective is 0.2%. The
resistors are sold in a lot of 100, with the
guarantee that all resistors are non-defective.
What is the probability that a given lot will
violate this guarantee?
Problem 11
Let P = 1% be the probability that a
certain type of light bulb will fail in 24
hours test. Find the probability that a sign
consisting of 100 such bulbs will burn 24
hours with no bulb failures.
Problem 13
Suppose that a test for extrasensory
perception consists of naming (in any
order) 3 card randomly drawn from a
deck of 13 cards. Find the probability that
by chance alone, the person will correctly
name (a) no cards, (b) 1 Card, (c) 2
Cards, and (d) 3 cards.
Ex. 6: Finding Binomial Probabilities
• A survey indicates that 41% of American
women consider reading as their favorite
leisure time activity. You randomly select
four women and ask them if reading is
their favorite leisure-time activity. Find the
probability that (1) exactly two of them
respond yes, (2) at least two of them
respond yes, and (3) fewer than two of
them respond yes.
Ex. 6: Finding Binomial Probabilities
• #1--Using n = 4, p = 0.41, q = 0.59 and x =2, the
probability that exactly two women will respond yes is:
P( 2)  4 C2 (0.41) 2 (0.59) 4 2 
4!
(0.41) 2 (0.59) 4 2 
( 4  2)!2!
24
(.1681)(.3481) 
4
6(.1681)(.3481)  .35109366
Calculator or look it up on pg. A10
Ex. 6: Finding Binomial Probabilities
•
#2--To find the probability that at least two women will respond yes, you can
find the sum of P(2), P(3), and P(4). Using n = 4, p = 0.41, q = 0.59 and x
=2, the probability that at least two women will respond yes is:
P ( 2)  4 C2 (0.41) 2 (0.59) 4  2  .351093
P (3)  4 C3 (0.41) 3 (0.59) 4 3  0.162653
P ( 4)  4 C4 (0.41) 4 (0.59) 4  4  0.028258
P ( x  2)  P ( 2)  P (3)  P ( 4)
 .351093 .162653 028258
 0.542
Calculator or look it up on pg. A10
Ex. 6: Finding Binomial Probabilities
•
#3--To find the probability that fewer than two women will respond yes, you
can find the sum of P(0) and P(1). Using n = 4, p = 0.41, q = 0.59 and x =2,
the probability that at least two women will respond yes is:
P(0) 4 C0 (0.41) 0 (0.59) 40  0.121174
P(1) 4 C1 (0.41)1 (0.59) 41  0.336822
P ( x  2)  P (0)  P (1)
 ..121174 .336822
 0.458
Calculator or look it up on pg. A10
Ex. 7: Constructing and Graphing a
Binomial Distribution
•
65% of American households subscribe to cable TV. You randomly select
six households and ask each if they subscribe to cable TV. Construct a
probability distribution for the random variable, x. Then graph the
distribution.
P (0)  6 C0 (0.65) 0 (0.35) 6 0  0.002
P (1)  6 C1 (0.65)1 (0.35) 6 1  0.020
P ( 2)  6 C2 (0.65) 2 (0.35) 6  2  0.095
P (3)  6 C3 (0.65) 3 (0.35) 6 3  0.235
P ( 4)  6 C4 (0.65) 4 (0.35) 6  4  0.328
P (5)  6 C5 (0.65) 5 (0.35) 6 5  0.244
P (6)  6 C6 (0.65) 6 (0.35) 6 6  0.075
Calculator or look it up on pg. A10
Ex. 7: Constructing and Graphing a
Binomial Distribution
•
65% of American households subscribe to cable TV. You randomly select
six households and ask each if they subscribe to cable TV. Construct a
probability distribution for the random variable, x. Then graph the
distribution.
x
P(x)
0
1
2
3
4
5
6
0.002
0.020
0.095
0.235
0.328
0.244
0.075
Because each probability is a relative frequency, you can graph the probability
using a relative frequency histogram as shown on the next slide.
Ex. 7: Constructing and Graphing a
Binomial Distribution
• Then graph the distribution.
x
P(x)
R
e
l
a
t
i
v
e
F
r
e
q
u
e
n
c
y
0
1
2
3
4
5
6
0.002
0.020
0.095
0.235
0.328
0.244
0.075
0.35
0.3
0.25
0.2
P(x)
0.15
0.1
0.05
0
0
1
2
3
4
Households
5
6
NOTE: that the
histogram is
skewed left. The
graph of a binomial
distribution with p >
.05 is skewed left,
while the graph of a
binomial distribution
with p < .05 is
skewed right. The
graph of a binomial
distribution with p =
.05 is symmetric.
Mean, Variance and Standard Deviation
• Although you can use the formulas
learned in 4.1 for mean, variance and
standard deviation of a probability
distribution, the properties of a binomial
distribution enable you to use much
simpler formulas. They are on the next
slide.
Quiz # 3
32 CE(B) – 12 NOV 2012
• Let P = 1% be the probability that a certain
type of light bulb will fail in 24 hours test.
Find the probability that a sign consisting
of 10 such bulbs will burn 24 hours with no
bulb failures.
(3 Marks)
• Write Probability Distribution Function for
Multinomial
and
Hypergeometric
distributions.
(2 Marks)