Question 1

The random walk theory of stock prices states that price movements in disjoint time periods are independent of each other.The probability that the value of a portfolio of stocks invested in companies listed on the TSE300 rises in any one year is 0.60.What is the probability that the portfolio's value moves in the same direction in both of the next 2 years?

Accepted Answer

The probability of the portfolio's value moving in the same direction in both of the next 2 years is 0.60 * 0.60 = 0.36.

Question 2

A hydro company finds that 80% of its customers pay a given monthly bill in full.Suppose 2 customers are chosen at random from the list of all customers.What is the probability that at least one of the two customers will pay their next monthly bill in full?

Accepted Answer

The probability that at least one of the two customers will pay their next monthly bill in full can be found by subtracting the probability that neither pays their bill in full from 1. The probability that a single customer does not pay in full is 1 - 0.80 = 0.20. Therefore, the probability that neither of the two customers pays in full is 0.20 * 0.20 = 0.04. Subtracting this from 1 gives 1 - 0.04 = 0.96.

Question 3

For a $1 ticket,a lottery offers the following prizes:
•1 prize of $50,000
•10 prizes of $10,000
•100 prizes of $100
•4000 prizes of 5 free ticket
If 250,000 tickets are sold,what is the expectation per ticket?

Accepted Answer

To find the expectation per ticket, we need to calculate the sum of the products of the probability of winning each prize and the amount of money won for that prize.

Probability of winning the $50,000 prize: 1/250,000
Probability of winning the $10,000 prize: 10/250,000
Probability of winning the $100 prize: 100/250,000
Probability of winning the 5 free tickets: 4,000/250,000

Expected value = (1/250,000)($50,000) + (10/250,000)($10,000) + (100/250,000)($100) + (4,000/250,000)(5) - 1
Expected value = 0.571 - 1
Expected value = -0.429

This means that on average, each ticket results in a loss of 42.9 cents. Therefore, the best choice is not to buy the ticket at all.

Question 4

Recent statistics show that the stock market (as measured by the TSE index)goes up in a day about 60% of the time and goes down the other 40% of the time.Statistics also show that if the market goes up one day,then it goes up the next day 70% of the time and down 30% of the time.But if the market goes down one day,then it goes up the next day 80% of the time and down 20% of the time.What is the probability that in three consecutive days,the market goes up for two of the days and down for one of the days?

Accepted Answer

The probability of the market going up two days and down one day in any order over three days can be calculated by considering all possible sequences of up (U) and down (D) days and their respective probabilities. The sequences are UUD, UDU, and DUU. 1. For UUD: The probability of UU is 0.6 * 0.7 (since there's a 60% chance it goes up on the first day and then a 70% chance it goes up again the next day). The probability of D given the previous day was U is 0.3 (since if it goes up one day, it goes down the next 30% of the time). So, the total probability for UUD is 0.6 * 0.7 * 0.3.2. For UDU: The probability of the first U is 0.6. The probability of D following a U is 0.3. The probability of U following a D is 0.8 (since if the market goes down one day, it goes up the next day 80% of the time). So, the total probability for UDU is 0.6 * 0.3 * 0.8.3. For DUU: The probability of the first D is 0.4 (since the market goes down 40% of the time). The probability of U following a D is 0.8. The probability of another U following the first U is 0.7. So, the total probability for DUU is 0.4 * 0.8 * 0.7.Adding these probabilities together gives the total probability of the market going up two days and down one day in three consecutive days:= (0.6 * 0.7 * 0.3) + (0.6 * 0.3 * 0.8) + (0.4 * 0.8 * 0.7)= 0.126 + 0.144 + 0.224= 0.494 or 49.4%.

Question 5

A 6-sided die has been weighted so that 5 or 6 comes up 3 times as often as 1 or 2.Also,3 or 4 comes up 2 times as often as 1 or 2.Player A wins $6 if 1 comes up,$5 if 2 comes up and $4 if 3 comes up.Player B wins $3 if 4 comes up,$2 if 5 comes up and $1 if 6 comes up.What is the value of E[A] - E[B],where E[A] is the expectation of winning for player A and E[B] is the expectation of winning for player B?

Accepted Answer

The expected value for player A is (1/6 * 6) + (1/6 * 5) + (1/6 * 4) = 5.The expected value for player B is (1/6 * 3) + (1/6 * 2) + (1/6 * 1) = 2.Therefore, E[A] - E[B] = 5 - 2 = 3.

Question 6

Recent statistics show that the stock market (as measured by the TSE index)goes up in a day about 60% of the time and goes down the other 40% of the time.Statistics also show that if the market goes up one day,then it goes up the next day 70% of the time and down 30% of the time.But if the market goes down one day,then it goes up the next day 80% of the time and down 20% of the time.What is the probability that in two consecutive days,the market goes in the opposite direction?

Accepted Answer

The answer of Recent statistics show that the stock market...

Question 7

You are given that 7.2% of the American population has Type O-negative blood.If 5 people are randomly selected to give blood,what is the probability that at least one of them has Type O-negative blood?

Accepted Answer

The answer of You are given that 7.2% of the...

Question 8

Recent statistics show that the stock market (as measured by the TSE index)goes up in a day about 60% of the time and goes down the other 40% of the time.Statistics also show that if the market goes up one day,then it goes up the next day 70% of the time and down 30% of the time.But if the market goes down one day,then it goes up the next day 80% of the time and down 20% of the time.What is the probability that in three consecutive days,the market goes in the same direction (either up or down)on each day?

Accepted Answer

To find the probability that the market goes in the same direction for three consecutive days, we need to consider two cases: either the market goes up for three days or the market goes down for three days.
Case 1: Market goes up for three days
The probability that the market goes up on any given day is 0.60. Therefore, the probability that the market goes up for three consecutive days is 0.60 * 0.70 * 0.70 = 0.294.
Case 2: Market goes down for three days
The probability that the market goes down on any given day is 0.40. Therefore, the probability that the market goes down for three consecutive days is 0.40 * 0.80 * 0.80 = 0.256.
The total probability that the market goes in the same direction for three consecutive days is the sum of the probabilities from each case: 0.294 + 0.256 = 0.55 or 31.0% (rounded to one decimal place).
Therefore, the best choice is B.

Question 9

Suppose that to get a job offer,you need to take 2 exams.The results of your exam are independent and you have 70% to pass each of them.If you fail both exams,you will have no chance to get the job.If you pass one of them,you will have 30% chance to get the job.If you pass both exams,you will have an 80% chance to get the job.What is the probability that you get the job?

Accepted Answer

To get the job, you need to pass at least one exam or pass both exams. The probability of passing both exams is 0.7 x 0.7 = 0.49. The probability of failing both exams is 0.3 x 0.3 = 0.09. The probability of passing one exam and failing the other is (0.7 x 0.3) + (0.3 x 0.7) = 0.42. Therefore, the probability of getting the job is the sum of the probabilities of passing at least one exam and passing both exams:
P(getting the job) = (0.49 x 0.8) + (0.42 x 0.3) = 0.518
So the best choice is C.

Question 10

You have two bags of marbles that have the following content:
Bag 1: 6 red,4 white Bag 2: 6 red,3 white
You reach into bag 1 (without looking),take out one marble and put it into bag 2.Then you reach into bag 2 and take out one marble.What is the probability that the marble chosen from bag 2 is red?

Accepted Answer

The probability of taking a red marble from Bag 1 and putting it into Bag 2 is 6/10 or 3/5. After transferring a red marble, Bag 2 would have 7 red and 3 white marbles, making the probability of drawing a red marble from Bag 2 equal to 7/11. If a white marble is transferred from Bag 1 to Bag 2, the probability of this event is 4/10 or 2/5, and Bag 2 would then have 6 red and 4 white marbles, making the probability of drawing a red marble from Bag 2 equal to 6/11. The total probability of drawing a red marble from Bag 2 is the sum of the probabilities of these two scenarios: (3/5)*(7/11) + (2/5)*(6/11) = 21/55 + 12/55 = 33/55, which simplifies to 0.60 or 60%.

Question 11

The probabilities that bond A,B,C and D will pay every one of their coupons along with the redemption value are 50%,70%,80% and 60% respectively.What is the probability that at least three of the bonds will fully meet their obligations?

Accepted Answer

The answer of The probabilities that bond A,B,C and D...

Question 12

While driving from his house to work each day,Jack Hammer passes 3 stoplights.If the probability of getting a red light is 1/3 for each of the 3 lights,what is the probability that Jack hits at least one red light?

Accepted Answer

The easiest way to solve this problem is to calculate the probability of the opposite event (not hitting any red lights) and subtracting that from 1. The probability of not hitting a red light at each stoplight is 2/3, so for all three lights, it's (2/3)^3 = 8/27. Therefore, the probability of hitting at least one red light is 1 - 8/27 = 19/27.

Question 13

A fair standard die is rolled until a six appears.If X is the number of rolls that are necessary.What is the probability that X ≤ 2?

Accepted Answer

The probability of getting a six in one roll is 1/6, and the probability of not getting a six in one roll is 5/6. The probability of getting a six on the first roll is 1/6, and the probability of not getting a six on the first roll is 5/6. The probability of getting a six on the second roll, given that a six did not appear on the first roll, is also 1/6. Therefore, the probability of getting a six in two rolls or less is the sum of the probabilities of getting a six on the first roll and getting a six on the second roll given that a six did not appear on the first roll, which is: 1/6 + (5/6)(1/6) = 11/36.

Question 14

Recent statistics show that the stock market (as measured by the TSE index)goes up in a day about 60% of the time and goes down the other 40% of the time.Statistics also show that if the market goes up one day,then it goes up the next day 70% of the time and down 30% of the time.But if the market goes down one day,then it goes up the next day 80% of the time and down 20% of the time.What is the probability that in two consecutive days,the market goes in the same direction (either up or down)on both days?

Accepted Answer

The answer of Recent statistics show that the stock market...

Question 15

A 6-sided die has been weighted so that 5 or 6 comes up 3 times as often as 1 or 2.Also,3 or 4 comes up 2 times as often as 1 or 2.Player A wins $6 if 1 comes up,$5 if 2 comes up and $4 if 3 comes up.Player B wins $3 if 4 comes up,$2 if 5 comes up and $1 if 6 comes up.Player B wins $3 if a 4 comes up,$2 if a 5 comes up and $1 if a 6 comes up.What is player B's expected winnings?

Accepted Answer

The answer of A 6-sided die has been weighted so...

Question 16

A fair coin is tossed 4 times.Event A is the probability that there are no heads and Event B is that probability that there are no tails.What is the probability of (Event A or Event B)?

Accepted Answer

The probability of getting no heads (all tails) in 4 tosses is (1/2)^4 = 0.0625. Similarly, the probability of getting no tails (all heads) is also 0.0625. Since these two events are mutually exclusive, the probability of either event A or event B occurring is 0.0625 + 0.0625 = 0.1250.

Question 17

A game show gives contestants 5 questions with 3 possible answers and awards $10 for every correct answer.If a contestant answers all 5 questions correctly,they win an extra $500.What is the probability of a contestant who guesses at every question will get at least 2 questions correct?

Accepted Answer

The probability of guessing at least 2 questions correctly out of 5 can be calculated by finding the complement of the probability of guessing 0 or 1 question correctly. The probability of guessing a question correctly is 1/3, and incorrectly is 2/3. Using the binomial probability formula, we calculate the probabilities for 0 and 1 correct guesses and subtract from 1.For 0 correct guesses: $P(X=0) = \binom{5}{0} (1/3)^0 (2/3)^5 = (2/3)^5$For 1 correct guess: $P(X=1) = \binom{5}{1} (1/3)^1 (2/3)^4 = 5 * (1/3) * (2/3)^4$Adding these probabilities gives the probability of getting at most 1 question right. Subtracting this sum from 1 gives the probability of getting at least 2 questions right, which is approximately 0.54.

Question 18

When John and Carl play a game,John wins,on average,2 out of every 5 games played.Today they agree to play until John wins one game or until three games have been played,whichever comes first.What is the probability that John wins one game today?

Accepted Answer

John's probability of winning any given game is 2/5, and the probability of losing any given game is 3/5. The probability that John wins at least one game today can be calculated by subtracting the probability of him losing all games (if they play three games) from 1. The probability of losing three games in a row is (3/5)^3 = 27/125. Therefore, the probability of winning at least one game is 1 - 27/125 = 98/125 = 0.784.

Question 19

A student has 3 exams.The probability that he does "well" on his first exam is 80%.However,his performance on the remaining two exams depends on how "well" he did on the previous exam.The probability of doing "well" on an exam is 95% if the previous exam was done "well",but only 50% if the previous was not done "well".What is the probability he does "well" on exactly 2 of his 3 exams?

Accepted Answer

The probability of doing well on the first exam is 80%. The probability of doing well on the second exam if the first exam was done well is 95%. The probability of doing well on the second exam if the first exam was not done well is 50%. The probability of doing well on the third exam if the second exam was done well is 95%. The probability of doing well on the third exam if the second exam was not done well is 50%.The probability of doing well on exactly 2 of the 3 exams is:P(well on first exam) * P(well on second exam | well on first exam) * P(not well on third exam | well on second exam) + P(well on first exam) * P(not well on second exam | well on first exam) * P(well on third exam | not well on second exam) = 0.8 * 0.95 * 0.05 + 0.8 * 0.05 * 0.95 = 0.153

Question 20

One day,20 people show up at a baseball diamond,13 males and 7 females.Since only 18 people are required to form two teams,two people are to be chosen at random to be the referees (umpires).What is the probability that one of them is male and one is female?

Accepted Answer

The answer of One day,20 people show up at a...

Quiz 8: Contingent Payments