MA 116 Statistics II Discussion
Jingjing Li
Boston University
September 14, 2022
Jingjing Li
MA 116 Statistics II Discussion
TF
▶ Name: Jingjing Li
▶ Office Hour: T 4:00 -5:00 PM, Th 9:30 - 10:30 AM, MCS
B27E
▶ Email: jli0203@bu.edu
▶ Content:
▶ Review concepts covered in the lecture
▶ Answer and Discuss questions related to homework
Jingjing Li
MA 116 Statistics II Discussion
Independent Events versus Disjoint Events
Disjoint events and independent events are different concepts.
Recall that two events are disjoint if they have no outcomes in
common, that is, if knowing that one of the events occurs, the
other event does not occur. Independence means that one event
occurring does not affect the probability of the other event
occurring.
Therefore, knowing two events are disjoint means that the events
are not independent.
Jingjing Li
MA 116 Statistics II Discussion
Check Independence
If E and F are two independent events, we have P(E |F ) = P(E ),
then we can derive
P(E and F ) = P(F ) · P(E |F ) = P(F ) · P(E )
To tell whether two events are independent:
▶ check whether P(E and F ) = P(E )P(F ) or
▶ check whether P(E |F ) = P(E ) or
▶ check whether P(F |E ) = P(F )
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MA 116 Statistics II Discussion
5.4.35
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MA 116 Statistics II Discussion
5.4.35 Solution
(a)
200
1
=
400
2
200
21
1
P(male|0 activ.) =
=
=
400
42
2
P(male) = P(male|0 activ.)
P(male) =
the events “male” and “0 activities” are independent.
(b)
1
200
=
400
2
71
≈ 0.651
P(female|5 + activ.) =
109
P(female) ̸= P(female|5 + activ.)
P(female) =
the events “female” and “5+ activities” are independent.
Jingjing Li
MA 116 Statistics II Discussion
5.4.35 Solution Cont.
(c) Yes, the events “1–2 activities” and “3–4 activities” are
mutually exclusive because the two events cannot occur at the
same time. That is P(1–2 activ. and 3–4 activ.) = 0
(d) No, the events “male” and “1–2 activities” are not mutually
exclusive because the two events can happen at the same
81
= 0.2025 ̸= 0
time. P(male and 1–2 activ.) = 400
Jingjing Li
MA 116 Statistics II Discussion
Permutation
The number of arrangements of r objects chosen from n objects, in
which
▶ 1. the n objects are distinct
▶ 2. repetition of objects is not allowed
▶ 3. order is important
n Pr
=
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n!
(n − r )!
MA 116 Statistics II Discussion
Combination
Number of combinations of n distinct objects taken r at a time
▶ 1. the n objects are distinct
▶ 2. repetition of objects is not allowed
▶ 3. order is not important
n Cr
=
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n!
r !(n − r )!
MA 116 Statistics II Discussion
Questions 5.5.49 & 55
Jingjing Li
MA 116 Statistics II Discussion
Solution
▶ 49.
25 P4
25!
25!
=
= 25 · 24 · 23 · 22
(25 − 4)!
21!
= 303, 600
=
▶ 50.
21 C9
21!
21!
=
9!(21 − 9)!
9!12!
21 · 20 · 19 · · · 13
=
9 · 8 · 7···1
= 293, 930
=
Jingjing Li
MA 116 Statistics II Discussion
Multiplication versus Addition
▶ Multiplication: If a task consists of a sequence of steps (that
is, you must finish all the steps so that you can finish the
task) in which the number ways of finishing the first step is p1
, the number ways of the second step is p2 , the number ways
of the third step is p3 , and so on, then the number ways of
finishing task successfully is p1 · p2 · p3 · · ·
▶ Addition: If a task can be finished via several methods (that
is, you just need to use either one of these methods in order
to finish the task), the number ways of using the first method
is q1 , the number ways of using the second method is q2 , the
number ways of using the third method is q3 and so on, then
the number ways of finishing the task is q1 + q2 + q3 + · · ·
Jingjing Li
MA 116 Statistics II Discussion
Section 5.5.62
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MA 116 Statistics II Discussion
Solution
Jingjing Li
MA 116 Statistics II Discussion
Binomial Probability Experiment
Criteria for a Binomial Probability Experiment
An experiment is said to be a binomial experiment if
1. The experiment is performed a fixed number of times. Each
repetition of the experiment is called a trial.
2. The trials are independent. This means the outcome of one
trial will not affect the outcome of the other trials.
3. For each trial, there are two mutually exclusive (or disjoint)
outcomes, success or failure.
4. The probability of success is fixed for each trial of the
experiment.
Jingjing Li
MA 116 Statistics II Discussion
Binomial Random Variable.
Let the random variable X be the number of successes in n trials
of a binomial experiment. Then X is called a binomial random
variable, that is, X ∼ Bin(n, p)
▶ Probability Mass Function(PMF) is given by
p(x) = P(X = x) =n Cx · p x · (1 − p)n−x , x = 0, 1, 2, . . . , n
▶ The Mean: µ = np.
▶ The Variance: σ 2 = npq.
p
√
▶ The SD: σ = np(1 − p) = npq.
Jingjing Li
MA 116 Statistics II Discussion
6.2.38
Allergy Sufferers Clarinex-D is a medication whose purpose is to
reduce the symptoms associated with a variety of allergies. In
clinical trials of Clarinex-D, 5% of the patients in the study
experienced insomnia as a side effect. A random sample of 20
Clarinex-D users is obtained, and the number of patients who
experienced insomnia is recorded.
(a) Find the probability that exactly 3 experienced insomnia as a
side effect.
(b) Find the probability that 3 or fewer experienced insomnia as a
side effect.
(c) Find the probability that between 1 and 4 patients inclusive,
experienced insomnia as a side effect.
(d) Would it be unusual to find 4 or more patients who
experienced insomnia as a side effect? Why?
Jingjing Li
MA 116 Statistics II Discussion
Hint
▶ (a) P(X = 3) = 20 C3 · (0.05)3 · (1 − 0.05)20−3
▶ (b) Using the cumulative binomial probability table,
P(X ≤ 3) = 0.9841
▶ (c) Two methods:
1. P(1 ≤ X ≤ 4) = P(1) + P(2) + P(3) + P(4)
2. P(1 ≤ X ≤ 4) = P(X ≤ 4) − P(X ≤ 0)
▶ (d)
P(X ≥ 4) = 1 − P(X < 4) = 1 − P(X ≤ 3)
Then compare the above probability with 0.05.
Jingjing Li
MA 116 Statistics II Discussion
Normal Random Variable
Normal probability distribution is determined by two parameters:
▶ location parameter: µ
▶ scale parameter: σ
Standard normal probability distribution: µ = 0, σ = 1.
Jingjing Li
MA 116 Statistics II Discussion
Two Objectives
1. Find the area under a normal curve give the normal random
variable value.
2. Find the value of a normal random variable given the
probability or area.
Jingjing Li
MA 116 Statistics II Discussion
Jingjing Li
MA 116 Statistics II Discussion
Find the area under a normal curve
For a Normal random variable X with general mean µ and
standard deviation σ,
P(a ≤ X ≤ b) = P(
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b−µ
a−µ
≤Z ≤
)
σ
σ
MA 116 Statistics II Discussion
Question 7.2.44
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MA 116 Statistics II Discussion
Find the Value of a Normal Random Variable
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MA 116 Statistics II Discussion
The Normal Approximation to the Binomial Probability
Distribution
If np(1 − p) ≥ 10, the binomial random variable X is
approximately normally distributed,
with mean µX = np and
p
standard deviation σX = np(1 − p)
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MA 116 Statistics II Discussion
Continuity Correction
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MA 116 Statistics II Discussion
Question 7.4.21
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MA 116 Statistics II Discussion
Solution
Jingjing Li
MA 116 Statistics II Discussion
Solution
Jingjing Li
MA 116 Statistics II Discussion