Question

1)

a) According to the survey in 3 (a), what fraction of those earning at least $70,000 per year are Democrats? (This is equivalent to asking for the probability that a person earning at least $70,000 per year is a Democrat. It is also equivalent to asking for the probability that a person is a Democrat, given that the person earns at least $70,000 per year. Symbolically, this is P ( D ∣ M ) ).

b) According to the survey in 3(a), what is the probability that a person selected at random fits at least one of these categories: that is the person is a Democrat or earns at least $70,000 per year. Symbolically, this is P ( D ∪ M ) . As in all the other parts of this problem, write your answer as a decimal, accurate to three decimal places.

c) Assume a survey of 500 students showed that 163 have a cat, 122 have a dog and 50 have both of these pets. How many of these 500 students have a cat, but not a dog? (Symbolically, this is n ( C ∩ D ¯ ) ).

d) A person has 19 shirts and 8 ties. How many different shirt and tie arrangements can he wear?

e) Assume A and B are mutually exclusive events, with P ( A ) = 0.23 and P ( B ) = 0.54. Find P ( A ∩ B ) .

2)

a) Assume A and B are mutually exclusive events, with P ( A ) = 0.26 and P ( B ) = 0.49. Find P ( A ∪ B ) . .

b) Assume A and B are independent events, with P ( A ) = 0.35 and P ( B ) = 0.47. Find P ( A ∩ B ) . Enter an answer accurate to three decimal places.

c) Assume A and B are independent events, with P ( A ) = 0.16 and P ( B ) = 0.46. Find P ( A ∪ B ) . Enter an answer accurate to three decimal places.

d)

Assume the universal set is S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , A = { 2 , 3 , 5 } and B = { 1 , 3 , 5 , 6 } . In this problem, write each answer as a single set. Be sure to use braces, { }, around the elements of each set. Write the numbers in each answer in order from least to greatest, separating them with commas, but don't use any spaces. What is

A ¯ ?

What is A ∪ B ?

What is A ∩ B ?

Answer 1

Q 1 (c):-

Total students =500

Let A be the Students have cats =163

Let B be the students have dogs =122

Let C be the students who have both dogs and cats =50

Using identity

n(A U B) = n(A) + n( B) - n(C) where C is define above

n(A U B) = total students

If we subtract C from A then we get the students who have only cats therefore

n ( A ∩ B ¯ ) = 163 - 50

= 113

Q1 (d) = He can select a shirt in 19 waya

And he can select tie in 8 ways

so total arrangments = 19*8

= 152

Q1(e) = As we know total probability is 1

P ( A ) = 0.23

P( B ) = 0.54.

As we know A and B are mutually exclusive then

P ( A ∩ B ) =0

Q2 (a) :-

P ( A ) = 0.26

P ( B ) = 0.49

As we know that P ( A ∪ B ) = P(A) + P(B)

= 0.26+0.49

= 0.75

Q2 (b) =

As given :

A and B are independent events

P ( A ) = 0.35

P ( B ) = 0.47

As we know A and B are independent then

P ( A ∩ B ) = P(A)*P(B)

= 0.35*0.47

=0.1645

= 0.165 (approx)

Q2(c)

A and B are independent events

P ( A ) = 0.16

P ( B ) = 0.46.

As we know A and B are independent then

P ( A ∩ B ) = P(A) *P(B)

Thenn using identity

P ( A ∪ B ) = P(A)+P(B)-P(A)*P(B)

P(A U B ) = 0.16+0.46-0.16*0.46

= 0.6936

=0.694(approx)

Q2(d)

Given

S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 }

A = { 2 , 3 , 5 }

B = { 1 , 3 , 5 , 6 }

Are the sets