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2.
HOMEWORK
Breakfast for Rupert
Rupert has either milk or cocoa to drink for breakfast with either
oatmeal or pancakes. If he drinks milk, then the probability that he is
having pancakes with the milk is 2/3. The probability that he drinks
cocoa is 1/5. If he drinks cocoa, the probability of him having
pancakes is 6/7.
a) Show the sample space of probabilities using a tree diagram or any
other method of your choice.
b) Find the probability that Rupert will have oatmeal with cocoa
tomorrow morning.
3.
Testing for independence ...
30% of seniors get the ﬂu every year. 50% of seniors get a ﬂu shot
annually. 10% of seniors who get the ﬂu shot also get the ﬂu. Are
getting a ﬂu shot and getting the ﬂu independent events?
HOMEWORK
P(shot) = 0.50
P(ﬂu) = 0.30
P(shot & ﬂu) = (0.50)(0.30) = 0.15
However
P(shot & ﬂu) = 0.10
4.
The probability that Gallant Fox will win the ﬁrst race is 2/5 and that
Nashau will win the second race is 1/3.
HOMEWORK
1. What is the probability that both horses will win their
respective races?
2. What is the probability that both horses will lose their
respective races?
3. What is the probability that at least one horse will win a race?
5.
The probability that Gallant Fox will win the ﬁrst race is 2/5 and that
Nashau will win the second race is 1/3.
HOMEWORK
3. What is the probability that at least one horse will win a race?
6.
Mutually Exclusive Events ...
Two events are mutually exclusive (or disjoint) if it is impossible
for them to occur together.
Formally, two events A and B are mutually exclusive if and
only if
Mutually Exclusive Not Mutually Exclusive
A A
B B
2
1
3 46
5
Examples:
1. Experiment: Rolling a die once
Sample space S = {1,2,3,4,5,6}
Events A = 'observe an odd number' = {1,3,5}
B = 'observe an even number' = {2,4,6}
A B = (the empty set), so A and B are mutually exclusive.
2. A subject in a study cannot be both male and female, nor can
they be aged 20 and 30. A subject could however be both male
and 20, or both female and 30.
7.
Example
Suppose we wish to ﬁnd the probability of drawing either a king or a
spade in a single draw from a pack of 52 playing cards.
We deﬁne the events A = 'draw a king' and B = 'draw a spade'
Since there are 4 kings in the pack and 13 spades, but 1 card is
both a king and a spade, we have:
P(A and B) = P(A B) P(A U B) = P(A) + P(B) - P(A B)
= P(A) * P(B)
= 4/52 + 13/52 - 1/52
= (4/52) * (13/52)
= 16/52
= 1/52
So, the probability of drawing either a king or a spade is 16/52 = 4/13.
B
A 2
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8.
and
Identify the events as:
Drop
Drag
not mutually exclusive
or
mutually exclusive
a. A bag contains four red and seven black marbles. The event is
randomly selecting a red marble from the bag, returning it to the bag,
and then randomly selecting another red marble from the bag.
not mutually exclusive
b. One card - a red card or a king - is randomly drawn from a
deck of cards. not mutually exclusive
c. A class president and a class treasurer are randomly selected
from a group of 16 students.
mutually exclusive
d. One card - a red king or a black queen - is randomly drawn
from a deck of cards.
mutually exclusive
e. Rolling two dice and getting an even sum or a double.
not mutually exclusive
9.
Probabilities involving quot;andquot; and quot;orquot; A.K.A quot;The Addition Rulequot;...
The addition rule is a result used to determine the probability that event
A or event B occurs or both occur.
The result is often written as follows, using set notation:
P(A or B) = P(A B) = P(A)+P(B) - P(A B)
where:
P(A) = probability that event A occurs
P(B) = probability that event B occurs
P(A U B) = probability that event A or event B occurs
P(A B) = probability that event A and event B both occur
P(A and B) = P(A B) = P(A)*P(B)
A
AB B
AB A B and
or
10.
Chad has arranged to meet his girlfriend, Stephanie, either in the
library or in the student lounge. The probability that he meets her
in the lounge is 1/3, and the probability that he meets her in the
library is 2/9.
HOMEWORK
a. What is the probability that he meets her in the library or
lounge?
b. What is the probability that he does not meet her at all?
11.
The probability that Tony will move to Winnipeg is 2/9, and the
probability that he will marry Angelina if he moves to Winnipeg is 9/20.
The probability that he will marry Angelina if he does not move to
Winnipeg is 1/20. Draw a tree diagram to show all outcomes.
HOMEWORK
1. What is the probability that Tony will move to Winnipeg and
marry Angelina?
2. What is the probability that Tony does not move to Winnipeg but
does marry Angelina?
3. What is the probability that Tony does not move to Winnipeg
and does not marry Angelina?
12.
(a) How many different 4 digit numbers are there in which all
the digits are different?
HOMEWORK
(b) If one of these numbers is randomly selected, what is the
probability it is odd?
(c) What is the probability it is divisable by 5?
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