Question

7. Consider the following scenario:

• Let P(C) = 0.2

• Let P(D) = 0.3

• Let P(C | D) = 0.4

• Part (a)
  Find P(C AND D).

• Part (b)
  Are C and D mutually exclusive? Why or why not?C and D are not mutually exclusive because
  P(C) + P(D) ≠ 1
  .C and D are mutually exclusive because they have different probabilities. C and D are not mutually exclusive because
  P(C AND D) ≠ 0
  .There is not enough information to determine if C and D are mutually exclusive.

• Part (c)
  Are C and D independent events? Why or why not?The events are not independent because the sum of the events is less than 1.The events are not independent because
  P(C) × P(D) ≠ P(C | D)
  . The events are not independent because
  P(C | D) ≠ P(C)
  .The events are independent because they are mutually exclusive.

• Part (d)
  Find P(D | C).

8. G and H are mutually exclusive events.

• P(G) = 0.5

• P(H) = 0.3

• Part (a)
  Explain why the following statement MUST be false:
  P(H | G) = 0.4.
  The events are mutually exclusive, which means they can be added together, and the sum is not 0.4.The statement is false because P(H | G) =

P(H)

P(G)

• = 0.6. To find conditional probability, divide
  P(G AND H) by P(H)
  , which gives 0.5.The events are mutually exclusive, which makes
  P(H AND G) = 0
  ; therefore,
  P(H | G) = 0.

• Part (b)
  Find
  P(H OR G).

• Part (c)
  Are G and H independent or dependent events? Explain

• G and H are dependent events because they are mutually exclusive.

• G and H are dependent events because

• P(G OR H) ≠ 1.

• G and H are independent events because they are mutually exclusive.

• There is not enough information to determine if G and H are independent or dependent events.

9.

Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3 percent speak Spanish.

• E = speaks English at home

• E' = speaks another language at home

• S = speaks Spanish at home

Finish each probability statement by matching the correct answer.

• Part (a)
  P(E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (b)
  P(E)
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (c)
  P(S and E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (d)
  P(S | E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

Answer 1

7(a) P( C and D) = P( C I D ) *P(D) = 0.4*0.3 =0.12

b) C and D are not mutually exclusive because P(C) +P(D) 1

Note : Two events C and D are said to be mutually exclusive if sum of their probabilities is equal to 1.

c)The events C and D are not indpendent because P( C I D) P(C)

Note : Two events C and D are said to be independent if probability of occurance of C given that D has occurred is same as probability of occurance of C. , that is C and D are said to be independent if P( C I D) = P(C) .

d) Using Baye's theorem

P( D I C) = P( C I D) *P( D) / P( C) =0.4*0.3/0.2 =0.6

8a) P( H I G) =0.4 is false

To find conditional probability divide P( H and G) by P(G)

The events are mutually exclusive which gives P( H and G) =0

Thus , P( H I G) = 0

b) P( H or G) = P( H) +P(G) - P( H and G)

= 0.3 +0.5- 0

= 0.8

c) G and H are dependent events as they are mutually exclusive

Note : Two events G and H are said to be independent if P( G and H ) = P( G) *P(H)

But P( G and H) =0 as they are mutually exclusive

P( G ) *P(H) = 0.3*0.5=0.15

Thus , we get P( G and H ) P( G) *P(H)

9a) P(E') = 55000000 / 281000000= 0.1957

b) P(E) = 1- P( E') =1-0.1957 = 0.8043

c) P( S and E') = P( S I E') *P( E') = 0.623*0.1957 =0.1219

d) P( S I E') = 0.6230