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Lecture 5

# MATA 67 Lecture 5: w5

Department
Mathematics
Course Code
MATA67H3
Professor
Richard Pancer
Lecture
5

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Sum and Product Rules
Exercise. Consider tossing a coin ﬁve times. What is the proba-
bility of getting the same result on the ﬁrst two tosses or the last
two tosses?
Solution.
Let Ebe the event that the ﬁrst two tosses are the same and F
be the event that the last two tosses are the same. Let n(E)be
the number of ways event Ecan occur.
n(E)=
n(F)=
n(S)=
Q. Is there any overlap in Eand F?I.e., is E\F=;?
A.
n(Eand F)=n(E\F)
=
=
1

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Now we can calculate the probability P(Eor F):
P(Eor F)=
=
=
=
Theorem (The Sum Rule) If Eand Fare events in an experi-
ment then the probability that Eor Foccurs is given by:
P(Eor F)=P(E)+P(F)P(Eand F)
Example. What is the probability when a pair of dice are rolled
that at least one die shows a 5 or the dice sum to 8?
Solution.
2
Let E5 be the event that at least one die shows a 5
E8 “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ the dice sums to 8
P(E5) = 11/36
P(E8) = 5/36
P(E 5 and E8) = 2/36
PANCERS SQUARE
P(E5 or E8) = P(E5) + P(E8) - P(E5 + E8)
= 11/36 + 5/36 - 2/36 = 14/36
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Exercise. Given a bag of 3 red marbles, 5 black marbles and 8
green marbles select one marble and then a second. What is the
probability that both are red?
Solution.
Q. What is the probability of the ﬁrst marble being red?
A.
Q. What is the probability of the second marble being red?
A.
Q. Probability of both marbles being red?
A.
When the probability of an event Edepends on a previous event
Fhappening we denote this probability as P(E|F).
3
Do not put back the ﬁrst marble
3/16
Not 3/16 Now to calculate the probability of the second marble being red given that the 1st marble is red 2/15
3/16 x 2/15 = 6/240
given that
Text
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