Download The Binomial Probability Distribution

Probability Distributions
Statistical Experiments and
Random Variables



Statistical Experiments – any process
by which measurements are obtained.
A quantitative variable x, is a random
variable if its value is determined by
the outcome of a random experiment.
Random variables can be discrete or
continuous.
Discrete vs. Continuous


Discrete random variables – can take
on only a countable or finite number
of values. (Countable)
Continuous random variables – can
take on countless values in an interval
on the real line. (Measurable)
Which measurement involves a
discrete random variable?
a). Determine the mass of a randomly-selected
penny
b). Assess customer satisfaction rated from 1
(completely satisfied) to 5 (completely
dissatisfied).
c). Find the rate of occurrence of a genetic
disorder in a given sample of persons.
d). Measure the percentage of light bulbs with
lifetimes less than 400 hours.
Discrete or Continuous?
Gas mileage(mpg) of all 2014 Toyota Prius’
 Number of songs on a seniors iPod
 1 mile time of a high school junior
 Amount of m&ms in a bag
 Height of a high school freshman

Probability distributions
An assignment of probabilities to the
specific values or a range of values for
a random variable.
Discrete Probability Distribution
1) Each value of the random variable
has an assigned probability.

2)
The sum of all the assigned
probabilities must equal 1.
Discrete Probability Distribution
Rolling a die

P(1) = .167, P(2) = .167, P(3) = .167…
0.18
0.16
0.14
1
2
3
4
5
6
0.12
0.1
0.08
0.06
0.04
0.02
0
# on the Die
Discrete Probability Distribution
Age of AP Statistics Students
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
15
16
17
Age of AP Statistics Students
18
19
Discrete Probability Distribution
We asked 200 adults how many cars
they own. The results are shown as a
probability distribution below.
# of cars
0
1
2
3
P(x)
.08
.72
.18
.02
1) What is the probability of someone owning 2 cars?
2) What is the probability of someone owning at least 1 car?
3) What is the probability of someone owning no more than 1
car?
Probability Distributions for
Continuous Random Variables
What would a histogram look like if we
graphed the weights (to the nearest
pound) of 100 newborn babies?
 What would it look like if we graphed the
same 100 newborn babies, however
rounded the weights to the nearest tenth?
 What about hundredth?

Probability Distributions for
Continuous Random Variables
The probability distribution is given as a
density curve.
 The function that defines this curve is
called the density function f(x)

Probability Distributions for
Continuous Random Variables

When the density is constant over an
interval, the probability distribution is
called a uniform distribution.
Probability Distribution Features


Since a probability distribution can be
thought of as a relative-frequency
distribution for a very large n, we can
find the mean and the standard
deviation.
When viewing the distribution in terms
of the population, use µ for the mean
and σ for the standard deviation.
Mean and Standard Deviation
of Discrete Probability Distribution

𝜇=

𝜎=
𝑥𝑃 𝑥
(Expected Value)
(𝑥 − 𝜇)2 𝑃(𝑥)
Find the mean and standard
deviation of the following…
x
0
1
2
3
4
5
6
7
8
9
10
p(x)
.002
.001
.002
.005
.02
.04
.17
.38
.25
.12
.01
µ = 7.16, 𝜎 = 1.57
1) What is the probability that a child
receives a score within 2 standard
deviations of the mean?
Sometimes we are interested in the
behavior of not just a random variable
itself, but a function of the variable.
Suppose we know the mean and standard
deviation of the number of gallons of propane a
random person orders in Old Lyme to be 318
and 42 respectively.
 A company is considering 2 different pricing
models:
Model 1: $3 per gallon
Model 2: service charge of $50 +$2.80 per gallon

Mean and Standard Deviation of
Linear Functions
Helps us understand the behavior of functions
of random variables.
 Mean of y = a + bx is



a + bµ𝑥
Standard deviation of y is
◦ 𝑏 𝜎𝑥
Which pricing model is better?
x = # of gallons of propane
𝜎𝑥 = 42 𝜇𝑥 = 318
 Model 1= $3.00x
 Model 2 = $50 + $2.80 per gallon
 We are interested in y(amount billed)
 Find the mean and standard deviation of
both models and compare them.
 𝜎1 = 126, 𝜇1 = 954
 𝜎2 = 117.60, 𝜇2 = 940.40

Mean and Standard Deviation of
Linear Combinations
𝜇𝑦 = 𝜇𝑥1 +𝜇𝑥2 + ⋯
 𝜎𝑦2 = 𝜎12 + 𝜎22 + ⋯
(must be independent)

Mean
Standard
Deviation
Mult. Choice
38
6
Free Response
30
7

If the free response is given twice as
much weight, what are the mean and s.d
of y?
Binomial Experiment (2 outcomes)
1)
2)
3)
There are a fixed number of trials.
This is denoted by n.
The n trials are independent and
repeated under identical conditions.
Each trial has two outcomes:
S = success
4)
F = failure
For each trial, the probability of
success, p, remains the same. Thus,
the probability of failure is 1 – p = q.
Binomial Probability

The central problem is to determine
the probability of x successes out of n
trials.
◦ Mark is an 80% free throw shooter. What
is the probability that he makes exactly 5
out of 10 free throws?
◦ What is the probability he makes exactly
10 out of 10?
◦ What is the probability he makes more
than half his free throws?
Probability of x successes


𝑃 𝑟 = 𝐶𝑛,𝑟 ∙ 𝑝𝑟 ∙ 𝑞 𝑛−𝑟
𝐶𝑛,𝑟 =
𝑛!
𝑟! 𝑛−𝑟 !
Find the probability of observing 6 successes in 10
trials if the probability of success is p = 0.4.
a). 0.111
b). 0.251
c). 0.0002
d). 0.022
Using the Binomial Table
1)
Locate the number of trials, n.
2)
Locate the number of successes, x.
3)
Follow that row to the right to the
corresponding p column.
Find the probability of observing 3
successes in 5 trials if p = 0.7
Finding the Probability of multiple
successes, r’s.

Find the probability of observing less than
three 3 successes in 5 trials if p = 0.7

P(r<3) = P(0) + P(1) + P(2)
Find the probability of observing less
than 3 successes in 5 trials if p = 0.7
Graphing a Binomial Distribution

Same as a relative frequency histogram
with r values on the horizontal axis.
Graph n=3, and p=.2
0.6
0.5
0.4
r=0
r=1
r=2
r=3
0.3
0.2
0.1
0
# of Successes
Mean and Standard Deviation
𝜇
= 𝑛𝑝
𝜎
= 𝑛𝑝𝑞
Graph n=3, and p=.2
0.6
0.5
0.4
r=0
r=1
r=2
r=3
0.3
0.2
0.1
0
# of Successes
𝜇 = .6
𝜎 = .6928
Think about it!
The graph of a distribution with a small
probability of success(p) would be skew left,
right, symmetrical, uniform or bimodal?
 How about a probability of success being 0.5?

The Normal Curve (Bell Curve)
Finding z scores

z score gives the number of standard
deviations between the x value and the
mean µ.
z=

𝑥−µ
σ
Always round to the hundredths.
Finding area (probability) using z
score and the standard NORMAL
DISTRIBUTION
Table 2 gives you the area to the left of
z.
 This is the probability of less than z.

Area to the right of z
Area Between two z values
Inverse Normal distribution
We have been working with finding an
area (probability), given a certain x or z
value.
 We can also find an x or z value given a
certain area (probability).

Left-tail case: the given area is to the
left of z

Look up the number A in the body of the table and
use the corresponding z value.
Right-tail case: the given area is to the
right of z

Look up the number 1 – A in the body of the table
and use the corresponding z value.
Center-tail case: the given area is
symmetric and centered above z = 0.

Look up the number (1 – A)/2 in the body of the table
and use the corresponding ±z value.
 Using
Table 3 in the Appendix, find the
range of z scores, centered about the
mean, that contain 70% of the probability.
 a).
–1.04 to 1.04
b). –2.17 to 2.17
 c).
–0.30 to 0.30
d). –0.52 to 0.52
Checking for Normality
Normal Probability Plot: the scatterplot of
(normal score, observed value)
 If the normal probability plot shows a
linear trend, then the data is
approximately normal.

Using the Correlation Coefficient to
Check Normality
Correlation Coefficient is obtained using
(normal score, observed value)
 If r is close to 1, then a linear relationship
should be represented.
 How close to 1 is close to 1?
 If r is less than the critical r for the
corresponding n, it is not reasonable to
assume a normal distribution
 If n is between two sample sizes, use the
larger n

Approximating Discrete/Binomial
Distributions using Normal Dist.
Approximating Discrete/Binomial
Distributions using Normal Dist.
How do we tell that a binomial
distribution is normal?
 If np ≥ 10 and nq ≤ 10, then the
distribution is approximately normal.
 Add or subtract .5 to your x-values and
compute the probability as you would a
normal distribution.


A Biologist found that the probability is
only 0.65 that a given Arctic Tern will
survive the migration from its summer
nesting are to its winter feeding groups. A
random sample of 500 Artic Terns were
branded at their summer nesting area.
Find the probability that between 310 and
340 of the branded Artic Terns will
survive the migration.