Question 1

Randall Realty has two rental properties available.The agent assesses a 0.6 probability that property A will be rented by the end of the month, and a probability of 0.8 that property B will be rented by the end of the month.She also believes there is a 0.5 probability that both properties will be rented by the end of the month.Is the renting of property A independent of the renting of property B?

Accepted Answer

For two events to be independent, the probability of one event occurring should not affect the probability of the other event occurring. The probability that A rents given B rents can be calculated using the formula for conditional probability: P(A rents given B rents) = P(A and B rent) / P(B rents) = 0.5 / 0.8 = 0.625. Since this is not equal to P(A rents), which is 0.6, the renting of property A is not independent of the renting of property B.

Question 2

In a recent survey of local businessmen and businesswomen, 40% of the respondents belong to the Rotary Club, 70% belong to the Chamber of Commerce.30% were members of both.What proportion of the respondents are members of at least one of the groups?

Accepted Answer

To find the proportion of respondents who are members of at least one of the groups, we use the principle of inclusion-exclusion. The formula is: $$ \text{At least one} = \text{Rotary Club} + \text{Chamber of Commerce} - \text{Both} $$Given that 40% belong to the Rotary Club, 70% belong to the Chamber of Commerce, and 30% belong to both, we have:$$ \text{At least one} = 40\% + 70\% - 30\% = 80\% $$To convert this percentage into the options provided (assuming they represent proportions in some form), 80% corresponds to 8 out of 10, or simply 8, which matches option E.

Question 3

When a machine is working properly only 4% of the units it produces are defective.When it slips out of adjustment, the rate jumps to 25% and the machine has to be shut down and adjusted.On 90% of the mornings when production restarts, the machine is in proper adjustment.You turn on the machine this morning and produce the first unit.If the first unit is defective, what is the probability that the machine is in adjustment?

Accepted Answer

The answer of When a machine is working properly only...

Question 4

As project manager at Gonzaga Contracting, you currently have a major project nearing completion, but are concerned about the delivery of the roofing materials needed to complete the project.Ace roofing, your main supplier, may need to first deliver materials to another of its clients.You put the chance of that happening at 30%.You believe that if Ace has to first deliver to its other client, there's a 60% chance that you will not get your delivery in time to complete your project on schedule.If Ace does not have to first deliver to its other client, you estimate that the chance that you will not get your delivery in time to finish on schedule falls to 10%.Determine the likelihood that you will not get the roofing delivery in time to finish on schedule.

Accepted Answer

The likelihood of not getting the roofing delivery in time can be calculated using the law of total probability. The probability of not getting the delivery on time if Ace has to deliver to another client first is 30% * 60% = 18%. The probability of not getting the delivery on time if Ace does not have to deliver to another client first is 70% * 10% = 7%. Adding these probabilities gives 18% + 7% = 25%.

Question 5

In a recent survey of local business owners, 40% of the respondents belong to the Rotary Club and 70% belong to the Chamber of Commerce.30% were members of both.Are the events "member of Chamber" and "member of Rotary" statistically independent?

Accepted Answer

Two events A and B are statistically independent if and only if P(A and B) = P(A) * P(B). Here, P(Chamber and Rotary) = 0.30, P(Chamber) = 0.70, and P(Rotary) = 0.40. If they were independent, P(Chamber and Rotary) should equal 0.70 * 0.40 = 0.28, but it's 0.30. Therefore, they are not independent, and the correct answer is that P(Chamber given Rotary) is not equal to P(Chamber), indicating dependence.

Question 6

As project manager at Gonzaga Contracting, you currently have a major project nearing completion, but are concerned about the delivery of the roofing materials needed to complete the project.Ace roofing, your main supplier, may need to first deliver materials to another of its clients.You put the chance of that happening at 30%.You believe that if Ace has to first deliver to its other client, there's a 60% chance that you will not get your delivery in time to complete your project on schedule.If Ace does not have to first deliver to its other client, you estimate that the chance that you will not get your delivery in time to finish on schedule falls to 10%.Suppose you now learn that Ace will not be able to deliver your roofing materials in time.How likely is that they had to first deliver to their other client?

Accepted Answer

The answer of As project manager at Gonzaga Contracting, you...

Question 7

Randall Realty has two rental properties available.The agent assesses a 0.6 probability that property A will be rented by the end of the month, and a probability of 0.8 that property B will be rented by the end of the month.She also believes there is a 0.5 probability that both properties will be rented by the end of the month.What is the probability that both properties fail to be rented by the end of the month?

Accepted Answer

The answer of Randall Realty has two rental properties available.The...

Question 8

When a machine is working properly only 4% of the units it produces are defective.When it slips out of adjustment, the rate jumps to 25% and the machine has to be shut down and adjusted.On 90% of the mornings when production restarts, the machine is in proper adjustment.You turn on the machine this morning and produce the first unit.If the first unit is not defective, what is the probability that the machine is out of adjustment?

Accepted Answer

Given that the first unit produced is not defective, we need to find the probability that the machine is out of adjustment. Let's denote:- $A$ as the event that the machine is out of adjustment,- $A'$ as the event that the machine is in proper adjustment,- $D'$ as the event that a unit is not defective.We are looking for $P(A|D')$, the probability that the machine is out of adjustment given that the first unit is not defective.Using Bayes' theorem:$$P(A|D') = \frac{P(D'|A)P(A)}{P(D')}$$Given:- $P(A) = 0.10$ (10% chance the machine is out of adjustment),- $P(A') = 0.90$ (90% chance the machine is in proper adjustment),- $P(D'|A) = 0.75$ (25% defective means 75% not defective if out of adjustment),- $P(D'|A') = 0.96$ (4% defective means 96% not defective if in proper adjustment).We also need $P(D')$, the overall probability of producing a non-defective unit, which can be found using the law of total probability:$$P(D') = P(D'|A)P(A) + P(D'|A')P(A')$$$$P(D') = 0.75 \times 0.10 + 0.96 \times 0.90$$$$P(D') = 0.075 + 0.864$$$$P(D') = 0.939$$Now, we can calculate $P(A|D')$:$$P(A|D') = \frac{0.75 \times 0.10}{0.939}$$$$P(A|D') = \frac{0.075}{0.939}$$$$P(A|D') \approx 0.080$$Therefore, the correct answer is C, which corresponds to a probability of approximately 0.080, or 8%.

Question 9

On any given morning, the probability that Alex catches his 8 AM train into the city is .5.What is the likelihood that Alex will miss the train on five consecutive mornings?

Accepted Answer

The probability of Alex missing his train on any given morning is 1 - 0.5 = 0.5. For five consecutive mornings, the probability is $0.5^5 = 0.03125$.

Question 10

When a machine is working properly only 4% of the units it produces are defective.When it slips out of adjustment, the rate jumps to 25% and the machine has to be shut down and adjusted.On 90% of the mornings when production restarts, the machine is in proper adjustment.You turn on the machine this morning and produce the first unit.How likely is it that the first unit is defective?

Accepted Answer

To find the probability that the first unit is defective, we need to consider both scenarios: when the machine is working properly and when it is not. The probability of the machine being in proper adjustment is 90%, and in this case, the probability of producing a defective unit is 4%. The probability of the machine being out of adjustment is 10%, with a 25% chance of producing a defective unit. Thus, the total probability of a defective unit is (0.9 * 0.04) + (0.1 * 0.25) = 0.036 + 0.025 = 0.061.

Question 11

In a recent survey of local business owners, 40% of the respondents belong to the Rotary Club, 70% belong to the Chamber of Commerce.30% were members of both.If a respondent is selected at random, and it turns out that that person is a member of the Chamber of Commerce, how likely is it that that person is also a member of the Rotary Club?

Accepted Answer

Given that a person is a member of the Chamber of Commerce, the probability they are also in the Rotary Club is found using the formula for conditional probability. With 30% in both, and 70% in the Chamber of Commerce, the probability is 30%/70% = 42.86%.

Question 12

In a recent survey of local business owners, 40% of the respondents belong to the Rotary Club, 70% belong to the Chamber of Commerce.30% were members of both.30% said that they read both.What proportion belong to the Chamber of Commerce but not the Rotary Club?

Accepted Answer

The answer of In a recent survey of local business...

Question 13

The probability that Omaha will be struck by a severe tornado during the year is 60%.The probability that Omaha will be struck by a severe blizzard is 30%.The likelihood that Omaha will be hit by both is put at 10%.What is the probability that at least one of the weather events occurs this year?

Accepted Answer

To find the probability of at least one event occurring, we can use the formula: P(A or B) = P(A) + P(B) - P(A and B). Substituting the given probabilities, we get: 0.6 + 0.3 - 0.1 = 0.8 or 80%.

Question 14

Randall Realty has two rental properties available.The agent assesses a 0.6 probability that property A will be rented by the end of the month, and a probability of 0.8 that property B will be rented by the end of the month.She also believes there is a 0.5 probability that both properties will be rented by the end of the month.What is the probability that property A is rented if property B is rented?

Accepted Answer

The probability that property A is rented given that property B is rented can be found using the formula for conditional probability: P(A|B) = P(A and B) / P(B). Given P(A and B) = 0.5 and P(B) = 0.8, P(A|B) = 0.5 / 0.8 = 0.625.

Question 15

According to a recent survey, 90% of all American CEO's favor a general lowering of tariffs between the US and the EU, while only 20% of EU CEO's favor such a move.At a conference of American and EU CEOs, 70% of those in attendance are American and the remaining 30% are from the EU.You choose a CEO at random and discover that the person you have chosen is a supporter of a general lowering of tariffs.What is the probability that the person is from the EU?

Accepted Answer

The answer of According to a recent survey, 90% of...

Question 16

The probability that Oklahoma will be struck by a severe tornado during the year is 60%.The probability that Oklahoma will be struck by a severe blizzard is 30%.The likelihood that Oklahoma will be hit by both is put at 10%.If Oklahoma is struck by a severe blizzard this year, how likely is it that it will also be struck by a severe tornado?

Accepted Answer

The probability that Oklahoma will be struck by both a severe tornado and a severe blizzard is 10%. Given that a severe blizzard occurs, the probability of also having a severe tornado is the probability of both events happening divided by the probability of the blizzard happening: 10% / 30% = 1/3 or 0.333, rounded to a single decimal place as per the options, it is 0.3 or 30%, which is not directly listed. The correct interpretation based on the options provided and rounding standards should be 0.5 (50%), indicating a misunderstanding in the calculation. The correct calculation should involve understanding conditional probability correctly. However, the explanation provided initially was incorrect due to a miscalculation. The correct approach involves using the formula for conditional probability, but the final interpretation of the answer choices seems to have been mistaken. Given the options and the typical rounding and interpretation of probabilities, none directly match the calculated 0.333. The question's phrasing suggests looking for the likelihood of a tornado given a blizzard, which should be calculated as P(Tornado and Blizzard)/P(Blizzard) = 10%/30% = 1/3. The error in the explanation was in the interpretation of the result in relation to the provided options. The correct answer, based on the calculation, should indeed be 0.333, but due to a mistake in interpreting the options, the explanation mistakenly suggested an answer of 0.5. The correct choice, based on the calculation provided, would be closest to option A if rounding and interpretation were correctly applied to match the options given.

Question 17

In a recent survey of local business owners, 40% of the respondents belong to the Rotary Club and 70% belong to the Chamber of Commerce.30% were members of both.Are the events "member of Chamber" and "member of Rotary" mutually exclusive?

Accepted Answer

Mutually exclusive events cannot occur at the same time. Since 30% of the respondents are members of both the Chamber of Commerce and the Rotary Club, the events are not mutually exclusive, indicating that P(Chamber and Rotary) is not equal to 0.

Question 18

In a recent survey of local businessmen and businesswomen, 40% of the respondents belong to the Rotary Club, 70% belong to the Chamber of Commerce.30% were members of both.What proportion belong to neither group?

Accepted Answer

20% belong to neither group. This is found by subtracting the sum of individual group memberships (40% + 70% = 110%) by the percentage of overlap (30%) and then subtracting from 100% to find those not in either group: 100% - (110% - 30%) = 20%.

Question 19

Randall Realty has two rental properties available.The agent assesses a 0.6 probability that property A will be rented by the end of the month, and a probability of 0.8 that property B will be rented by the end of the month.She also believes there is a 0.5 probability that both properties will be rented by the end of the month.What is the probability that at least one of the properties is rented by the end of the month

Accepted Answer

The probability that at least one of the properties is rented by the end of the month can be calculated using the formula: P(A or B) = P(A) + P(B) - P(A and B). Substituting the given probabilities, we get: 0.6 + 0.8 - 0.5 = 0.9 or 90%.

Question 20

The probability that Oklahoma will be struck by a severe tornado during the year is 60%.The probability that Oklahoma will be struck by a severe blizzard is 30%.The likelihood that Oklahoma will be hit by both is put at 10%.What is the probability that neither weather event will occur this year?

Accepted Answer

The answer of The probability that Oklahoma will be struck...

Quiz 4: Probability