Let A and B be two events in a sample space S. If P(A1Bc) =.10,
P(Ac1B) =.40 and P(Ac1Bc) =.10,
a. Illustrate the events A1Bc, Ac1B and Ac1Bc on a van
Venn-diagram
b. Check for the independence of events A and B.
c. Find P(A1B*AUB).
Let A, B and C be mutually independent events of a probability
space (Ω, F, P), such that P(A) = P(B) = P(C) = 1 4 . Compute P((Ac
∩ Bc ) ∪ C). b) [4 points] Suppose that in a bicycle race, there
are 19 professional cyclists, that are divided in a random manner
into two groups. One group contains 10 people and the other group
has 9 people. What is the probability that two particular people,
let’s say...
Sample Spaces
1. Suppose S is a uniform sample space with N elements. If E is
any possible come and ω is the probability function for S evaluate
ω(e).
2. Define a probability function on the set A = {1, 2, 3} such
that A is not a uniform sample space.
3. Given the sample space B = {a, b, c} and probability function
ω on B. If ω(a) = 0.3, ω({a, b}) = 0.8 then find ω(b) and ω(c)....
Describe the difference between the probability of two mutually
exclusive events, two complementary events, and two events that are
not mutually exclusive. Give examples of each.
If two events are independent how do we calculate the
and probability, P(E and F), of the two
events?
(As a side note: this "and" probability, P(E and F), is called
the joint probability of Events E and F. Likewise,
the probability of an individual event, like P(E), is called the
marginal probability of Event E.)
1.A company is analyzing two mutually exclusive projects, E and
F, whose cash flows are shown below:
Years
0
1
2
3
4
Cash Flow E
-$1,100
$900
$350
$50
$10
Cash Flow F
-$1,100
$0
$300
$400
$850
The company's cost of capital is 12 percent, and it can get an
unlimited amount of capital at that cost. What is the regular IRR
(not MIRR) of the better project? (Hint: Note that the better
project may or may not...
Suppose f : X → S and F ⊆ P(S). Show, f −1 (∪A∈F A) = ∪A∈F f −1
(A) f −1 (∩A∈F A) = ∩A∈F f −1 (A)
Show, if A, B ⊆ X, then f(A ∩ B) ⊆ f(A) ∩ f(B). Give an example,
if possible, where strict inclusion holds.
Show, if C ⊆ X, then f −1 (f(C)) ⊇ C. Give an example, if
possible, where strict inclusion holds.
MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE
Determine whether the following pair of events are mutually
exclusive.
1) A card is drawn from a deck.
C={It is a King} D={It is a heart}.
2) Two dice are rolled.
G={The sum of dice is 8} H={One die shows a 6}
3) A family has three children.
K={First born is a boy} L={The family has children of both
sexes}
Use the addition rule to find the following probabilities.
1) A die is...