binomial distributions. binomial experiments probability experiments for which the results of each...
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Binomial Distributions
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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.
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Binomial Probability Distribution A Fixed Number of Observations (trials), n
15 tosses of a coin; 20 patients; 1000 people surveyed
A Binary Outcome 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
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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.
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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.
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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.
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Example
A certain surgical procedure has an
85% chance of success. A doctor
performs the procedure on eight
patients. The random variable
represents the number of successful
surgeries.
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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 = 8p = 0.85q = 1 – 0.85 = 0.15x = 0, 1, 2, 3, 4, 5, 6, 7, 8
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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.
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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.
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Binomial Probability Formula
xnxxnxx
n qpxxn
nqpCxP
!)!(
!)(
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Example: Binomial Probabilities
• A six sided die is rolled 3 times. Find the
probability of rolling exactly one 6.
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Roll 1 Roll 2 Roll 3 Frequency # 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
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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:
xnxxnxxn qp
xxn
nqpCxP
!)!(
!)(
Binomial Probability Formula
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Example: Binomial Probabilities
347.072
25
)216
25(3
)36
25)(
6
1(3
)6
5)(
6
1(3
)6
5()
6
1(
!1)!13(
!3)1(
2
131
P
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Binomial Probability Distribution
• By listing the possible values of x with
the corresponding probability of each,
we can construct a Binomial
Probability Distribution.
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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.
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Constructing a Binomial Distribution
044.0)64.0()36.0()0( 7007 CP
173.0)64.0()36.0()1( 6117 CP
292.0)64.0()36.0()2( 5227 CP
274.0)64.0()36.0()3( 4337 CP
154.0)64.0()36.0()4( 3447 CP
052.0)64.0()36.0()5( 2557 CP
010.0)64.0()36.0()6( 1667 CP
001.0)64.0()36.0()7( 0777 CP
x P(x)
0 0.044
1 0.173
2 0.292
3 0.274
4 0.154
5 0.052
6 0.010
7 0.001
P(x) = 1
Notice all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.
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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
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Population Parameters of a Binomial Distribution
Mean: = np
Variance: 2 = npq
Standard Deviation: = √npq
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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
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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.
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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?
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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?
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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.
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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?
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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?
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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.
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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.
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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.
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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:
35109366.)3481)(.1681(.6
)3481)(.1681(.4
24
)59.0()41.0(!2)!24(
!4
)59.0()41.0()2(
242
24224
CP
Calculator or look it up on pg. A10
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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:
028258.0)59.0()41.0()4(
162653.0)59.0()41.0()3(
351093.)59.0()41.0()2(
44444
34334
24224
CP
CP
CP
542.0
028258162653.351093.
)4()3()2()2(
PPPxP
Calculator or look it up on pg. A10
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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:
336822.0)59.0()41.0()1(
121174.0)59.0()41.0()0(141
14
04004
CP
CP
458.0
336822.121174..
)1()0()2(
PPxP
Calculator or look it up on pg. A10
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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.
Calculator or look it up on pg. A10
075.0)35.0()65.0()6(
244.0)35.0()65.0()5(
328.0)35.0()65.0()4(
235.0)35.0()65.0()3(
095.0)35.0()65.0()2(
020.0)35.0()65.0()1(
002.0)35.0()65.0()0(
66666
56556
46446
36336
26226
16116
06006
CP
CP
CP
CP
CP
CP
CP
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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.
Because each probability is a relative frequency, you can graph the probability using a relative frequency histogram as shown on the next slide.
x 0 1 2 3 4 5 6
P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075
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Ex. 7: Constructing and Graphing a Binomial Distribution• Then graph the distribution.
x 0 1 2 3 4 5 6
P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6
P(x)
Relative Frequency
Households
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.
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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.
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Quiz # 332 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)