ec41f10_hw2_sol - Kata Bognar [email protected] Economics 41...

Kata Bognar
[email protected]
Economics 41
Statistics for Economists
UCLA
Fall 2010
Homework Assignment 2.
- suggested solutions by Gonca Senel -
Conditional Probability.
1. A fair coin is flipped twice. Let the events A = “the first toss is heads” , B = “the second toss is heads”
and C = “at least one toss is heads” . Determine the following probabilities.
Answer: The outcome space for this random experiment is { HH, HT, TH, TT } .
• A = { HH, HT }
• B = { HH, TH }
• C = { HH, HT, TH }
• A ∩ B = { HH }
• B ∩ C = { HH, TH }
• A ∩ C = { HH, HT }
• C ∩ B
0
= { HT } .
Each outcome is equally likely therefore,
(a) P ( B ) =
1
2
(b) P ( B | A ) =
1 / 4
2 / 4
=
1
2
(c) P ( B | C ) =
2 / 4
3 / 4
=
2
3
(d) P ( C ) =
3
4
(e) P ( C | A ) =
2 / 4
2 / 4
= 1
(f) P ( C | B
0
) =
1 / 4
2 / 4
=
1
2
2. Two independent events are:
(a) always mutually exclusive events
(b) never mutually exclusive events
(c) always complementary events
(d) always subjective events
3. An independent research team inspects 500 batteries manufactured by two companies for being
good or defective. The following table gives the two-way classification of these 500 batteries.
Good Defective
Company A 150 50
Company B 260 40
(a) The team selects one battery randomly from these 500 batteries. Find the probability that
this battery:
Answer: Define the following events:
• A = “the battery selected is manufactured by company A” ,
• B = “the battery selected is manufactured by company B” ,
• G = “the battery selected is good” and
• D = “the battery selected is defective” .
1

i. is manufactured by company B: Answer: P ( B ) =
300
500
ii. is defective: Answer: P ( D ) =
90
500
iii. is good given that it is manufactured by company B: Answer: P ( G | B ) =
260
300
iv. is manufactured by company A given that it is defective: Answer: P ( A | D ) =
50
90
.