Question

Let A = {−3,−1,1,3,5} and B = {−1,0,1,2,3,4,5}. What
are the following cardinalities?

(a) |A×B|

(b) |A−B|

(c) |A ∪B|

(d) |P(B)|

(e) |(A −B)×(B −A)|

Answer #1

Let P[A] = P[B] = 1/3 and P[A ∩ B] = 1/10.
Find the following:
(a) P[A ∪ Bc ]
(b) P[Ac ∩ B]
(c) P[Ac ∪ Bc ]
Now, let P[A] = 0.38 and P[A ∪ B] = 0.84.
(d) For what value of P[B] are A and B mutually exclusive?
(e) For what value of P[B] are A and B independent?

Let X ∼ B(5, 0.3). Determine:
(b) P(X < 2);
(c) P(X > 2);
Let X ∼ P0(2.1). Determine:
(a) P(X = 5);
(b) P(X < 3);
(c) P(X ≥ 3);
(d) E(X);
(e) var(X).

Appendix B Problem 3
Let Ω = {1,2,...,100} and let A,B and C be the following subsets
of Ω.
A = {positive even numbers which are at most 100}
B = {two-digit numbers where the digit 5 appears}
C = {positive integer multiples of 3 which are at most 100}
D = {two-digit numbers such that the sum of the digits is 10}
List the elements of each of the following sets:
a) B\A
b) A∩B∩Cc
c) ((A\D)∪B)∩(C ∩D)

Let lst1 = '(a b c d e),
lst2 = '(1 2 3 4 5). The
proc function takes a list as input and
returns the middle element of the list. What will be the following
expression evaluate to:
(append (cons (first lst1) '())
(cons (cdddr lst1) '())
(cons (proc lst1) '())
(cons (proc lst2) '())
(cons (fifth lst1) '())
)
a.
'(a d e c 3 e)
b.
'(a d e c 3 d)
c.
'(a (d e)...

Let S = {a, b, c, d, e, f} with P(b) = 0.21, P(c) = 0.11, P(d) =
0.11, P(e) = 0.18, and P(f) = 0.19. Let E = {b, c, f} and F = {b,
d, e, f}. Find P(a), P(E), and P(F).

3. Answer the following question about cardinality: (a) A = {a},
what is |A|? (b) B = {a, a}, what is |B|? (c) C = {{a}}, what is
|C|? (d) D = {a, {a}}, what is |D|? (e) E = {∅, {∅}, {∅, {∅}}},
what is |E|? (f) X = {{a, {a, b}, {c}}, {b, {a, c}, {d, e, f}}},
what is |X|?

Let A,B ∈ M3(R) with |A| = 4 and |B| = −3.
Evaluate (a) |AB|, (b) |5A|, (c) |BT|, (d) |A−1|, (e) |A3|, (f)
|ATA|, (g) |B−1AB|.

Let P(A) = 0.4, P(A| B) = 0.6 and P(B | A) = 0.75. a. P(A ∩ B)
b. P(B) c. P(A ∪ B) d. P(A ∩ B) e. P(B | A)

2. Let A be of dimension 3 × 4, B be of dimension 3 × 3, C be of
dimension 2 × 3, and D be of dimension 3 × 2. Determine which of
the following calculations are possible. If they are, give the
dimensions. If not, explain why not.
(a) (BA)C
b) CD + BA

38. Let E and F be events with P(E) = .3, P(F) = .6, and P(E ∪
F) = .7. Find (a) P(E ∩ F) (b) P(E|F) (c) P(F|E) (d) P(Ec ∩ F) (e)
P(Ec |F)
answers: a. .2 b. .5 c. .67 d. .4 e. Please show work
how to get these answers and include venn diagram thank you

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