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BIOL499A
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Blaine Mullins
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Chapter 14-16

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Biology (Biological Sciences)

BIOL499A

Blaine Mullins

Winter

Description

Chapters 14 - 16: Probability
In these chapters, we will talk about
Sample space,
Events,
Probability for some events,
Complement of an event,
The union of two events,
The intersection of two events,
Mutually exclusive or disjoint events,
Probability rules,
Conditional probabilities,
Independence of two events,
Random variables,
Probability distribution of a discrete random variable,
Expected value or mean of a discrete random variable.
Definition: A probability experiment is any action for which an outcome
cannot be predicted with certainty.
HO2 2 0
Toss a coin: H = Head or T = Tail
Flip a die: 1, 2, 3, 4, 5, 6
Choose a student and record his/her weight: Definition: The set of all possible outcomes is called thesample space.
Example 1: Find the sample space for the following probability
experiments:
(a) Toss a coin:,T}
(b) Flip a die:2,3,4,5,6}
(c) Toss a coin twice:
S={(H,H), (H,T), (T,H), (T,T)} ={HH,HT,TH,TT}
(d) Flip a die twice:
(1,1), (1, 2),(1,3), (1, 4),1,5), (1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
S
(6,),(6,2),(6,3),(6,4),(6,5,(6,6)
(e) Toss a coin until we get a head: Definition: An event is a subset of outcomes from the sample space. An
individual outcome from the sample space is called a simple event or an
elementary event.
e.g.: Flip a die:= {1,2,3,4,5,6}
A = {2}
B = Odd numbers = {1,3,5}
C = Even numbers = {2,4,6}
D = Prime numbers = {2,3,5}
Definition: The probability of an event is a numerical value that represents
the proportion of times that the even t is expected to occur when the
experiment is repeated under identical conditions.
e.g.: Toss a coin S ={H, T}
P({H}) = 1/2 and P(T) = 1/2
For an equally likely model:
Number of elements in A
P(A ) Number of elements in Sor any set A Example 2: Toss a coin twice and record the outcome heaH) or tailT( )
for each toss. Find the probability of the following events:
S = {HH, HT, TH,TT}
A = Exactly one head = {HT, TH}
B = No head = {TT}
C = At least one head = {HT,TH, HH}
D = At most one head = {TT, HT, TH}
E = Both heads = {HH}
Example 3: Flip a die twice. Find the probability that the sum of two
numbers is seven.
Solution:
A = sum is 7 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} Example 4: A restaurant serves three types of pasta (Spaghetti, Rigatoni,
and Fettuccine) with one of two sauces (Tomato or White). A customer will
order a pasta dish. How should the owner go about determining the
probability that a customer will order
A = A spaghetti, B = A white-sauce dish.
Example 5: An insurance company checked its records for a recent year and
found that of 12299 automobile insurance policies in effect, 2073 made a
claim. Among insured derivers under age 25, there were 1053 claims out of
5192 policies. Find the probability that
(a) an insured driver will make a claim.
(b) an insured driver under 25 years will make a claim. Note: For any probability model, the following two conditions must hold:
a) For any event A, we have 0 ▯ P(A) ▯ 1
PE() 1
b)
E is simple
Therefore, for examples:
a) It is possible to have a coin with P({H}) = 0.7 and P(T) = 0.3.
But:
b) It is impossible to have a coin with P({H}) = 0.7 and P(T) = 0.4.
OR
c) It is impossible to have a coin with P({H}) = 1.2 and P(T) = ▯0.2.
Example 6: Is it possible to have a die with
1 2
P(1) P) (5)Pad (2) (4) (6) ?
9 9
Solution:
a) All probabilities are between zero and 1
b) P(1)P)P (3) (4) (5) (6) 1 2+1 2 1+2 + + + =1
9 9 9 9 9 9
Then
A = Odd numbers = {1,3,5}
B = Even numbers = {2,4,6}
C = Prime numbers = {2,3,5} Definition: The intersection of tAandvB, denotedA B, is
the set of all elementary outcomesAthatBare inoccurrence of
A Bmeans that bAtandBoccur.
e.g.: Flip a die:,3,4,5,6}
A n numbers 2,4,6
B {2}
Prime numbers ,3,5
A Even numbers 2,4,6
AC {}
C Odd numbers 1,5
Definition: Two evAnandB are mutually exclusive or disjoint when
they cannot occur simultaneously. Definition: Two eventsA and B are independent when the occurrence (or
the nonoccurrence) ofA does not affect the probability of the occurrence of
B and vice versa. It can be shown that two events A and B are
independent, if and only if, we hP(A B) P(A)P(B).
Example 7: Toss a coin twice. Let
A = First coin is head. B = Second coin is head, and C = At least one head
(a) Are A and B independent? How about disjoint?
(b) Are A and C independent? How about disjoint?
Solution: S = {HH,HT,TH,TT}
(a)
A HH HT,
A BHH { }
HTH,
P( ) PAPB )()
(b)
A HH ,HT
A TCHH { , }
C H,TH ,
PA( )( C PA )(P)C Definition: The union of two events A and B , denoted byA B , is the set
of all elementary outcomes that are iA B , or both. The occurrence of A B
means that either A or B or both occur.
e.g.: Flip a die: S={1,2,3,4,5,6}
A Even numbers 2,4,6
AB {2,4,6}{2,3,5} {2,4,6,2,3,5} {2,3,4,5,6}
B Prime numbers ,3,5
A Even numbers 2,4,6
C Odd numbers ,3,5 AC ,6 1,3,5 {2,4,6,1,3,5} {1,2,3,4,5,6}
Probability Rule: For any two AvandB, we have:
P(A B) P(A) P(B) P(A B)
Note thaP(A B) P(A) P(B) and only if A and B are disjoint.
Example 8: Flip a die twice. Let A = Sum of two numbers is seven, and B
= First die is 2. Find probaA B.y of
Solution:
A = sum is 7 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}
B = First is 2 = {(2,1),(2,2), 2,3), (2,4),(2,5), (2,6)}
, , , ,1
AB 2,1 , 2,2 ,(2,3 , 2,4 , , 25, 2,6
, 6,
, , 2,, (2,3),,4 , , 2,6 Example 9: An urn contains 10 balls labeled 1,2,,10 . Suppose one ball is
drawn at random from the urn and consider the following events:
A = The number is an even number,
B = The number is among the first five numbers,
C = The number is among the last three numbers.
(a) Find the following probabilities:
P(A B) , P(AC) P(B C) P(B C)
(b) Are A and B independent? How about disjoint?
(c) Are B and C independent? How about disjoint? C
Definition: The complement of an event A, denoted by A Aor

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