Lesson test: Conditional probability

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Conditional probability

1.[2p]
What does the denominator in $P (A | B) = P (A \cap B) / P (B)$ do?

It rescales the surviving outcomes so they again sum to one

Correct

It removes the outcomes in $A$ that are not in $B$

The answer is: It rescales the surviving outcomes so they again sum to one

It corrects for $A$ and $B$ being disjoint

The answer is: It rescales the surviving outcomes so they again sum to one

It converts a frequency into a degree of belief

The answer is: It rescales the surviving outcomes so they again sum to one

2.[3p]
Two fair dice are rolled and the total is at least nine. What is the probability that at least one die shows a six?

CorrectNot quite: 0.7

3.[2p]
Two disjoint events with positive probability are independent.

True

The answer is: False

False

Correct

4.[3p]
Three events are pairwise independent. What follows about the whole family?

Nothing: pairwise independence does not imply that all three multiply together

Correct

They are independent, since independence is a pairwise property

The answer is: Nothing: pairwise independence does not imply that all three multiply together

They must be disjoint

The answer is: Nothing: pairwise independence does not imply that all three multiply together

Their probabilities must sum to one

The answer is: Nothing: pairwise independence does not imply that all three multiply together

5.[2p]
A component fails with probability $0.02$ , independently of the others. A system of four such components works only if all four survive. What is the probability that the system works? Give the answer to four decimal places.

CorrectNot quite: 0.9224

6.[3p]
Three machines make $50$ , $30$ and $20$ percent of the output at defect rates of $1$ , $2$ and $3$ percent. What fraction of the whole output is defective? Give the answer as a decimal.

CorrectNot quite: 0.017

7.[2p]
In the three-door game, the host who knows where the car is opens a losing door and offers a switch. What is the probability that switching wins?

CorrectNot quite: 0.667

8.[3p]
Why does switching help in the three-door game?

The host never opens the car or your door, so his forced choice concentrates the odds on the door he left

Correct

Because two doors remain and each is equally likely

The answer is: The host never opens the car or your door, so his forced choice concentrates the odds on the door he left

Because the car is moved after your first pick

The answer is: The host never opens the car or your door, so his forced choice concentrates the odds on the door he left

Because your first pick was more likely to be wrong than any later pick

The answer is: The host never opens the car or your door, so his forced choice concentrates the odds on the door he left

9.[2p]
Match each expression to what it means.

$P (A \cap B)$
$P (A | B)$
$P (A) P (B)$
$P (A \cap B) = 0$

equals $P (A \cap B)$ when independent
A happens given B did
both happen
the two are disjoint

Show the answer

$P (A \cap B)$ : both happen $P (A | B)$ : A happens given B did $P (A) P (B)$ : equals $P (A \cap B)$ when independent $P (A \cap B) = 0$ : the two are disjoint

1.[2p] What does the denominator in P(A|B)=P(A∩B)/P(B) do?

2.[3p] Two fair dice are rolled and the total is at least nine. What is the probability that at least one die shows a six?

3.[2p] Two disjoint events with positive probability are independent.

4.[3p] Three events are pairwise independent. What follows about the whole family?

5.[2p] A component fails with probability 0.02, independently of the others. A system of four such components works only if all four survive. What is the probability that the system works? Give the answer to four decimal places.

6.[3p] Three machines make 50, 30 and 20 percent of the output at defect rates of 1, 2 and 3 percent. What fraction of the whole output is defective? Give the answer as a decimal.

7.[2p] In the three-door game, the host who knows where the car is opens a losing door and offers a switch. What is the probability that switching wins?

8.[3p] Why does switching help in the three-door game?

9.[2p] Match each expression to what it means.

1.[2p]
What does the denominator in $P (A | B) = P (A \cap B) / P (B)$ do?

2.[3p]
Two fair dice are rolled and the total is at least nine. What is the probability that at least one die shows a six?

3.[2p]
Two disjoint events with positive probability are independent.

4.[3p]
Three events are pairwise independent. What follows about the whole family?

5.[2p]
A component fails with probability $0.02$ , independently of the others. A system of four such components works only if all four survive. What is the probability that the system works? Give the answer to four decimal places.

6.[3p]
Three machines make $50$ , $30$ and $20$ percent of the output at defect rates of $1$ , $2$ and $3$ percent. What fraction of the whole output is defective? Give the answer as a decimal.

7.[2p]
In the three-door game, the host who knows where the car is opens a losing door and offers a switch. What is the probability that switching wins?

8.[3p]
Why does switching help in the three-door game?

9.[2p]
Match each expression to what it means.