Transcript MATH 1410/4.1 and 4.2 pp

Discrete Probability
Distributions
Probability Distributions
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A random variable x represents a
numerical value associated with each
outcome of a probability experiment.
It is DISCRETE if it has a finite number of
possible outcomes.
 It is CONTINUOUS if it has an
uncountable number of possible outcomes
(represented by an interval)
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The number of books in a university
library.
The amount of snow (in inches) that fell
in Nome, Alaska last winter.
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A list of each possible value and its
probability. Must satisfy 2 conditions:
1.
 2.
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0 < P(x) < 1
ΣP(x) = 1
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The # of games played in the World
Series from 1903 to 2009
# of games 4 5 6 7
played
Frequency 20 23 23 36
8
3
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The # of 911 calls received per hour.
X
0
P(x) 0.1
1
2
3
4
5
6
7
0.10 0.26 0.33 0.18 0.06 0.03 0.03
Notation: E(x)
Expected value represents what you would
expect to happen over thousands of trials.
SAME as the MEAN!!!
E(x) = µ = Σ[x·P(x)]
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If x is the net gain to a player in a game of
chance, then E(X) is usually negative. This
value gives the average amount per game the
player can expect to lose.
A charity organization is selling $5 raffle
tickets. First prize is a trip to Mexico valued
at $3450, second prize is a spa package valued
at $750. The remaining 20 prizes are $25 gas
cards. The number of tickets sold is 6000.
Binomial Distributions
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CONDITIONS:
1. there are a fixed number of independent
trials (n = # of trials)
2. Two possible outcomes for each trial,
Success or Failure.
3. Probability of Success is the same for each
trial. p = P(Success) and q = P(Failure)
4. random variable x = # of successful trials
If binomial, ID ‘success’, find n, p, q; list
possible values of x. If not binomial,
explain why.
From past records, a clothing store finds
that 26% of people who enter the store
will make a purchase. During a onehour period, 18 people enter the store.
The random variable represents the # of
people who do NOT make a purchase.
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To find the probability of exactly x
number of successful trials:
 P(x)
= nC x ·
x
p
·
n
–x
q
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A surgical technique is performed on 7
patients. You are told there is a 70% chance of
success. Find the probability that the surgery
is successful for
A) exactly 5 patients
B) at least 5 patients
C) less than 5 patients
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Construct a probability distribution,
then find mean, variance, and standard
deviation for the following:
One in four adults claims to have no
trouble sleeping at night. You randomly
select 5 adults and ask them if they have
trouble sleeping at night.
Consider “success” to mean no trouble
sleeping.
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