Question 1

Suppose that a marksman hits the bull's-eye 1 time in 40 shots. If the next 4 shots are independent, find the probability that all of the next 4 shots hit the bull's-eye. ​

Accepted Answer

The answer of Suppose that a marksman hits the bull's-eye...

Question 2

An urn contains 7 red, 3 white, and 6 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that one ball is white W and one is red R. ​

Accepted Answer

The answer of An urn contains 7 red, 3 white,...

Question 3

Three balls are drawn, without replacement, from a bag containing 9 red balls and 4 white balls. Construct a probability tree to determine the probability that three white W balls are drawn. ​

Accepted Answer

The answer of Three balls are drawn, without replacement, from...

Question 4

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, one ball is red R and one is white W. ​

Accepted Answer

The answer of A bag contains 5 white balls and...

Question 5

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn, with replacement, the first ball is red R or the second is white W. ​

Accepted Answer

The answer of A bag contains 5 white balls and...

Question 6

Assuming there are 365 different birthdays, find the probability that at least two of nine people chosen at random will have same birthdays. Round your answer to five decimal places. ​

Accepted Answer

The answer of Assuming there are 365 different birthdays, find...

Question 7

Three balls are drawn, without replacement, from a bag containing 8 red balls and 6 white balls. Construct a probability tree to determine the probability that the third ball drawn is red R. ​

Accepted Answer

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Question 8

An urn contains 4 red, 7 white, and 9 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that both balls are white W. ​

Accepted Answer

The answer of An urn contains 4 red, 7 white,...

Question 9

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye. ​

Accepted Answer

The answer of Suppose that a marksman hits the bull's-eye...

Question 10

A bag contains 7 white balls and 8 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, the first ball is red R and the second is white W. ​

Accepted Answer

The answer of A bag contains 7 white balls and...

Question 11

Lactose intolerance affects about 10% of non-Hispanic white Americans, 50% of Hispanic Americans, and 70% of African, Asian, and Native Americans. Seventy-four percent of U.S. residents are non-Hispanic whites, 10% of them are Hispanic, and 11% are African, Asian, or Native American. If a person is selected from this group of people, what is the probability, to four decimal places, that the person will be lactose intolerant? ​

Accepted Answer

The answer of Lactose intolerance affects about 10% of non-Hispanic...

Question 12

A bag contains 9 white balls W and 9 red balls R. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that at least one white W ball is drawn. ​

Accepted Answer

The answer of A bag contains 9 white balls W...

Question 13

Each of urns I and II has 5 red balls, 2 white balls, and 3 green balls. Urn III has 5 red balls, 3 white balls, and 2 green balls. An urn is selected at random and a ball is drawn. If the ball is green G, find the probability that urn III was selected. ​

Accepted Answer

We need to find the probability that urn III was selected given that a green ball was drawn. We can use Bayes' theorem:

P(urn III | G) = P(G | urn III) * P(urn III) / P(G)

P(G | urn III) is the probability of drawing a green ball from urn III, which is 2/10 = 1/5.
P(urn III) is the probability of selecting urn III, which is 1/3 since there are three urns.
P(G) is the overall probability of drawing a green ball, which is the weighted average of the probabilities of drawing a green ball from each urn:

P(G) = P(G | urn I) * P(urn I) + P(G | urn II) * P(urn II) + P(G | urn III) * P(urn III)

= (3/10 * 1/3) + (3/10 * 1/3) + (2/10 * 1/3)

= 1/5

Plugging these values into Bayes' theorem, we get:

P(urn III | G) = (1/5) * (1/3) / (1/5)

= 1/3

So the probability that urn III was selected given that a green ball was drawn is 1/3. The correct answer choice is C.

Question 14

A bag contains 7 white balls and 3 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn, with replacement, both balls are red R. ​

Accepted Answer

The answer of A bag contains 7 white balls and...

Question 15

A bag contains 8 white balls and 3 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that exactly one white W ball is drawn. ​

Accepted Answer

The answer of A bag contains 8 white balls and...

Question 16

A bag contains 5 coins, of which 4 are fair; the remaining coin has a head on both sides. If a coin is selected at random from the bag and tossed three times, construct a probability tree to determine the probability that heads will occur at least twice. ​

Accepted Answer

The probability tree can be constructed as follows:
First Branch:
P(Fair) = 4/5, P(Biased) = 1/5
Second Branch:
P(Head) = 1/2 for Fair, P(Head) = 1 for Biased
P(Tail) = 1/2 for Fair, P(Tail) = 0 for Biased
Third Branch:
P(Head) = 1/2 for Fair, P(Head) = 1 for Biased
P(Tail) = 1/2 for Fair, P(Tail) = 0 for Biased
The probability of getting at least two heads can be calculated by adding the probabilities of getting 2 heads and 3 heads.
For Fair coin:
P(2 heads) = (4/5)(1/2)(1/2)(1/2) = 1/16
P(3 heads) = (4/5)(1/2)(1/2)(1/2) = 1/16
For Biased coin:
P(2 heads) = (1/5)(1)(1)(1/2) = 1/10
P(3 heads) = (1/5)(1)(1)(1) = 1/5
Using the Law of Total Probability, the total probability of getting at least two heads can be calculated as:
P(at least two heads) = P(Fair)P(at least two heads|Fair) + P(Biased)P(at least two heads|Biased)
= (4/5)(1/16+1/16) + (1/5)(1/10+1/5) = 11/40
Therefore, the correct answer is C.

Question 17

An urn contains 8 red, 9 white, and 2 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that at least one ball drawn is black B. ​

Accepted Answer

The answer of An urn contains 8 red, 9 white,...

Question 18

Three balls are drawn, without replacement, from a bag containing 9 red balls and 7 white balls. Construct a probability tree to determine the probability that two white W balls and one red R ball are drawn in any order. ​

Accepted Answer

First, draw the probability tree with three levels representing the three draws. At each level, branch into two probabilities for drawing a red or a white ball. At the third level, label each branch with the corresponding order of the three draws (e.g. RWW for a red ball on first draw and two white balls on second and third draw). 
Next, calculate the probabilities for each possible order of drawing two white balls and one red ball by multiplying the probabilities along the corresponding branches and adding them up:
P(RWW) = P(R) × P(W) × P(W|R) = (9/16) × (7/15) × (7/14) = 27/112
P(WRW) = P(W) × P(R) × P(W|R) = (7/16) × (9/15) × (7/14) = 27/112
P(WWR) = P(W) × P(W|R) × P(R) = (7/16) × (9/15) × (5/14) = 15/112
P(Total) = P(RWW) + P(WRW) + P(WWR) = 69/224
Therefore, the probability of drawing two white balls and one red ball in any order is 69/224, which corresponds to choice B.

Question 19

Suppose a box contains 2 defective transistors and 12 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is good G and the second transistor is defective D? ​

Accepted Answer

The answer of Suppose a box contains 2 defective transistors...

Question 20

A bag contains 9 white balls and 5 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that all three balls are white W. ​

Accepted Answer

The answer of A bag contains 9 white balls and...

Quiz 8: Introduction to Probability

Suppose that a marksman hits the bull's-eye 1 time in 40 shots. If the next 4 shots are independent, find the probability that all of the next 4 shots hit the bull's-eye. ​

An urn contains 7 red, 3 white, and 6 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that one ball is white W and one is red R. ​

Three balls are drawn, without replacement, from a bag containing 9 red balls and 4 white balls. Construct a probability tree to determine the probability that three white W balls are drawn. ​

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, one ball is red R and one is white W. ​

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn, with replacement, the first ball is red R or the second is white W. ​

Assuming there are 365 different birthdays, find the probability that at least two of nine people chosen at random will have same birthdays. Round your answer to five decimal places. ​

Three balls are drawn, without replacement, from a bag containing 8 red balls and 6 white balls. Construct a probability tree to determine the probability that the third ball drawn is red R. ​

An urn contains 4 red, 7 white, and 9 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that both balls are white W. ​

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye. ​

A bag contains 7 white balls and 8 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, the first ball is red R and the second is white W. ​

A bag contains 9 white balls W and 9 red balls R. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that at least one white W ball is drawn. ​

Each of urns I and II has 5 red balls, 2 white balls, and 3 green balls. Urn III has 5 red balls, 3 white balls, and 2 green balls. An urn is selected at random and a ball is drawn. If the ball is green G, find the probability that urn III was selected. ​

A bag contains 7 white balls and 3 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn, with replacement, both balls are red R. ​

A bag contains 8 white balls and 3 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that exactly one white W ball is drawn. ​

A bag contains 5 coins, of which 4 are fair; the remaining coin has a head on both sides. If a coin is selected at random from the bag and tossed three times, construct a probability tree to determine the probability that heads will occur at least twice. ​

An urn contains 8 red, 9 white, and 2 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that at least one ball drawn is black B. ​

Three balls are drawn, without replacement, from a bag containing 9 red balls and 7 white balls. Construct a probability tree to determine the probability that two white W balls and one red R ball are drawn in any order. ​

Suppose a box contains 2 defective transistors and 12 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is good G and the second transistor is defective D? ​

A bag contains 9 white balls and 5 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that all three balls are white W. ​

Suppose that a marksman hits the bull's-eye 1 time in 40 shots. If the next 4 shots are independent, find the probability that all of the next 4 shots hit the bull's-eye.

An urn contains 7 red, 3 white, and 6 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that one ball is white W and one is red R.

Three balls are drawn, without replacement, from a bag containing 9 red balls and 4 white balls. Construct a probability tree to determine the probability that three white W balls are drawn.

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, one ball is red R and one is white W.

A bag contains 5 white balls and 5 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn, with replacement, the first ball is red R or the second is white W.

Assuming there are 365 different birthdays, find the probability that at least two of nine people chosen at random will have same birthdays. Round your answer to five decimal places.

Three balls are drawn, without replacement, from a bag containing 8 red balls and 6 white balls. Construct a probability tree to determine the probability that the third ball drawn is red R.

An urn contains 4 red, 7 white, and 9 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that both balls are white W.

Suppose that a marksman hits the bull's-eye 1 time in 10 shots. If the next 4 shots are independent, find the probability that exactly two of the next 4 shots hit the bull's-eye.

A bag contains 7 white balls and 8 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn,with replacement, the first ball is red R and the second is white W.

A bag contains 9 white balls W and 9 red balls R. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that at least one white W ball is drawn.

Each of urns I and II has 5 red balls, 2 white balls, and 3 green balls. Urn III has 5 red balls, 3 white balls, and 2 green balls. An urn is selected at random and a ball is drawn. If the ball is green G, find the probability that urn III was selected.

A bag contains 7 white balls and 3 red balls. Construct a probability tree to determine the probability that if 2 balls are drawn, with replacement, both balls are red R.

A bag contains 8 white balls and 3 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that exactly one white W ball is drawn.

A bag contains 5 coins, of which 4 are fair; the remaining coin has a head on both sides. If a coin is selected at random from the bag and tossed three times, construct a probability tree to determine the probability that heads will occur at least twice.

An urn contains 8 red, 9 white, and 2 black balls. One ball is drawn from the urn, it is replaced, and a second ball is drawn. Construct a probability tree to determine the probability that at least one ball drawn is black B.

Three balls are drawn, without replacement, from a bag containing 9 red balls and 7 white balls. Construct a probability tree to determine the probability that two white W balls and one red R ball are drawn in any order.

Suppose a box contains 2 defective transistors and 12 good transistors. If 2 transistors are drawn from the box without replacement, what is the probability that the first transistor is good G and the second transistor is defective D?

A bag contains 9 white balls and 5 red balls. Three balls are drawn, without replacement, from the bag. Construct a probability tree to determine the probability that all three balls are white W.