Question 1

A coin is tossed  times. What is the probability that the coin will land  on the  toss given that  were thrown on the first toss?

Accepted Answer

This problem can be solved using conditional probability formula: P(A|B) = P(A intersection B) / P(B). Let A be the event of getting heads on the 11th toss and B be the event of getting heads on the 1st toss. We are given that B has occurred. So, P(B) = 1/2. Now, if B has occurred, then we need to find the probability of A occurring. That means, out of the remaining 10 tosses, we need to get all tails followed by a head. The probability of getting tails is 1/2 and the probability of getting heads is also 1/2. So, the probability of getting all tails followed by a head is (1/2)^11. Hence, P(A intersection B) = (1/2)^11. So, P(A|B) = P(A intersection B) / P(B) = (1/2)^11 / (1/2) = (1/2)^10 = 11eab448_e7e5_e729_acdb_61fdc8b0bad6_TB6027_11. Therefore, the answer is C.

Question 2

A nationwide survey conducted by the National Cancer Society revealed the following information: Of 10,000 people surveyed, 3,300 were "heavy coffee drinkers" and 160 had cancer of the pancreas. Of those who had cancer of the pancreas, 142 were heavy coffee drinkers. Using the data in this survey, determine whether the events "being a heavy coffee drinker" and "having cancer of the pancreas" are independent events.

Accepted Answer

If the events were independent, then the probability of being a heavy coffee drinker and having cancer of the pancreas would be the product of their individual probabilities. However, in this case, the probability of having cancer of the pancreas is higher among heavy coffee drinkers (142/3300 = 0.043) than among non-heavy coffee drinkers (18/6700 = 0.003), indicating that the events are not independent.

Question 3

At a certain medical school,  of the students are from a minority group. Of those students who belong to a minority group,  are black. What is the probability that a student selected at random from this medical school is black? What is the probability that a student selected at random from this medical school is black if it is known that the student is a member of a minority group?

Accepted Answer

The answer of At a certain medical school,  of...

Question 4

An automobile manufacturer obtains the microprocessor used to regulate fuel consumption in its automobiles from three microelectronic firms: A, B, and C. The quality-control department of the company has determined that 2% of the microprocessors produced by firm A are defective, 1.5% of those produced by firm B are defective, and 3% of those produced by the firm C are defective. Firms A, B, and C supply 40%, 30%, and 30%, respectively, of the microprocessors used by the company. What is the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor?

Accepted Answer

To find the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor, we need to find the weighted average of the probability of a defective microprocessor from each of the three firms, using their respective market share as weights. 
P(defective) = (0.02 x 0.40) + (0.015 x 0.30) + (0.03 x 0.30) 
P(defective) = 0.008 + 0.0045 + 0.009 
P(defective) = 0.0215 
Therefore, the probability of a randomly selected automobile manufactured by the company having a defective microprocessor is 0.0215 or 2.15%. The answer is option C.

Question 5

Three cards are drawn without replacement from a well-shuffled deck of 36 playing cards. What is the probability that the first card drawn is a heart?

Accepted Answer

The answer of Three cards are drawn without replacement from...

Question 6

If A and B are independent events,  , and  , find  .

Accepted Answer

The answer of If A and B are independent events,...

Question 7

A pair of fair dice is cast. Let E denote the event that the number landing uppermost on the first die is a 6 and let F denote the event that the sum of the numbers falling uppermost is 12. Determine whether E and F are independent events.

Accepted Answer

E and F are dependent events. If E occurs, then the maximum value possible for F is 6+6=12. So, P(F|E)≠P(F), which shows that E and F are not independent events.

Question 8

Four black balls and five white balls are placed in an urn. Two balls are then drawn in succession. What is the probability that the second ball drawn is a white ball if the second ball is drawn without replacing the first?

Accepted Answer

The answer of Four black balls and five white balls...

Question 9

A card is drawn from a well-shuffled deck of 36 playing cards. Let E denote the event that the card drawn is red and let F denote the event that the card drawn is a hearts. Determine whether E and F are dependent events.

Accepted Answer

E and F are dependent events because drawing a red card increases the probability of drawing a heart, since half of the red cards in the deck are hearts. Therefore, the occurrence of event E affects the probability of event F.

Question 10

A pair of fair dice is cast. Let E denote the event that the number falling uppermost in the first die is 5 and let F denote the event that the sum of the numbers falling uppermost is 8. Compute  . Are E and F dependent events?

Accepted Answer

We need to find the probability of both events E and F occurring:
P(E and F) = P(E) x P(F|E)

Probability of E = number 5 on first die = 1/6
Probability of F given that E has occurred: 
There are 5 options for the number on the first die, but only one of these (3) will give a sum of 8 with the other die. Therefore, P(F|E) = 1/36
P(E and F) = (1/6) x (1/36) = 1/216

To check for independence:
P(E) = 1/6
P(F) = 5/36
P(E and F) = 1/216

If E and F are independent events, then P(E and F) = P(E) x P(F). However, we can see above that P(E and F) is not equal to P(E) x P(F), so E and F are dependent events.

Therefore, the correct answer is D: 11eab448_e7e3_eb38_acdb_2f7a20263be8_TB6027_11, yes.

Question 11

A pair of fair dice is cast. What is the probability that the sum of the numbers falling uppermost is less than 5, if it is known that one of the numbers is a 2? ​

Accepted Answer

If one of the numbers is a 2, then there are only 6 possible outcomes:
{(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)}

Out of these 6 possible outcomes, only (2,1) and (2,2) have a sum that is less than 5.

Therefore, the probability of getting a sum less than 5, given that one of the numbers is a 2, is 2/6 or 1/3.

Choice A is the only one that matches this probability (1/3).

Question 12

The probability that a battery will last 14 hr or more is .50, and the probability that it will last 25 hr or more is .12. Given that a battery has lasted 14 hr, find the probability that it will last 25 hr or more. ​

Accepted Answer

The answer of The probability that a battery will last...

Question 13

Two cards are drawn without replacement from a well-shuffled deck of 52 playing cards. What is the probability that the second card drawn is a heart given that the first card drawn was not a heart? What is the probability that the second card drawn is a heart given that the first card drawn was a heart? ​

Accepted Answer

The answer of Two cards are drawn without replacement from...

Question 14

A tax specialist has estimated the probability that a tax return selected at random will be audited is .06. Furthermore, he estimates the probability that an audited return will result in additional assessments being levied on the taxpayer is .50. What is the probability that a tax return selected at random will result in additional assessments being levied on the taxpayer?

Accepted Answer

The probability that a tax return will be audited and result in additional assessments is the product of the probability of being audited (0.06) and the probability of additional assessments given an audit (0.50), which is 0.06 * 0.50 = 0.03.

Question 15

The admissions office of a private university released the following admission data for the preceding academic year: From a pool of 4,400 male applicants, 30% were accepted by the university, and of these, 50% subsequently enrolled. Additionally, from a pool of 3,600 female applicants, 40% were accepted by the university, and of these, 30% subsequently enrolled. What is the probability that a student who applies for admission will be accepted by the university and subsequently will enroll?

Accepted Answer

The probability that a male applicant is accepted and enrolls is 0.30 * 0.50 = 0.15. The probability that a female applicant is accepted and enrolls is 0.40 * 0.30 = 0.12. The overall probability is (4400/8000)*0.15 + (3600/8000)*0.12 = 0.1365.

Question 16

The proprietor of Cunningham's Hardware Store has decided to install floodlights on the premises as a measure against vandalism and theft. If the probability is 0.01 that a certain brand of floodlight will burn out within a year, find the minimum number of floodlights that must be installed to ensure that the probability that at least one of them will remain functional within the year is at least 0.99999. (Assume that the floodlights operate independently.)

Accepted Answer

The answer of The proprietor of Cunningham's Hardware Store has...

Question 17

A box contains two defective Christmas tree lights that have been inadvertently mixed with seven nondefective lights. If the lights are selected one at a time without replacement and tested until both defective lights are found, what is the probability that both defective lights will be found after three trials? ​

Accepted Answer

The answer of A box contains two defective Christmas tree...

Question 18

Determine whether the given events A and B are independent.  ,  , and  . ​

Accepted Answer

Without any information regarding the nature of events A and B, we cannot definitively determine their independence. The given events do not provide any information to suggest that A and B are independent, so we must assume that they are dependent.

Question 19

In a survey of 1500 eligible voters selected at random, it was found that 150 had a college degree. Additionally, it was found that 90% of those who had a college degree voted in the last presidential election, whereas 45% of the people who did not have a college degree voted in last presidential election. Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random had a college degree and voted in the last presidential election. ​

Accepted Answer

The probability that an eligible voter selected at random had a college degree and voted in the last presidential election is calculated as follows: Out of 1500 eligible voters, 150 had a college degree. 90% of these 150 college-educated voters voted in the last election, which is 0.9 * 150 = 135 voters. Therefore, the probability is 135 out of 1500, which simplifies to 0.09.

Question 20

If A and B are independent events,  , and  , find  .

Accepted Answer

Since A and B are independent events, P(A and B) = P(A) * P(B) 
Given:
P(A) = 0.7 
P(B) = 0.3 
We need to find P(A and not B)
P(A and not B) = P(A) * P(not B) 
= P(A) * (1 - P(B)) 
= 0.7 * (1 - 0.3) 
= 0.7 * 0.7 
= 0.49

Therefore, the answer is C) 11eab448_e7e3_27db_acdb_0dc5d5968fe8_TB6027_11.

Quiz 7: Probability

A coin is tossed times. What is the probability that the coin will land on the toss given that were thrown on the first toss?

Three cards are drawn without replacement from a well-shuffled deck of 36 playing cards. What is the probability that the first card drawn is a heart?

If A and B are independent events, , and , find .

A pair of fair dice is cast. Let E denote the event that the number landing uppermost on the first die is a 6 and let F denote the event that the sum of the numbers falling uppermost is 12. Determine whether E and F are independent events.

Four black balls and five white balls are placed in an urn. Two balls are then drawn in succession. What is the probability that the second ball drawn is a white ball if the second ball is drawn without replacing the first?

A card is drawn from a well-shuffled deck of 36 playing cards. Let E denote the event that the card drawn is red and let F denote the event that the card drawn is a hearts. Determine whether E and F are dependent events.

A pair of fair dice is cast. Let E denote the event that the number falling uppermost in the first die is 5 and let F denote the event that the sum of the numbers falling uppermost is 8. Compute . Are E and F dependent events?

A pair of fair dice is cast. What is the probability that the sum of the numbers falling uppermost is less than 5, if it is known that one of the numbers is a 2? ​

The probability that a battery will last 14 hr or more is .50, and the probability that it will last 25 hr or more is .12. Given that a battery has lasted 14 hr, find the probability that it will last 25 hr or more. ​

Determine whether the given events A and B are independent. , , and . ​

If A and B are independent events, , and , find .

A pair of fair dice is cast. What is the probability that the sum of the numbers falling uppermost is less than 5, if it is known that one of the numbers is a 2?

The probability that a battery will last 14 hr or more is .50, and the probability that it will last 25 hr or more is .12. Given that a battery has lasted 14 hr, find the probability that it will last 25 hr or more.

Determine whether the given events A and B are independent. , , and .