Lesson test: Outcomes and events

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Outcomes and events

1.[2p]
Why is the sample space for two dice usually taken as the $36$ ordered pairs rather than the $11$ possible totals?

Only the pairs are interchangeable, so only they can be counted as equally likely

Correct

The totals do not cover every possible result

The answer is: Only the pairs are interchangeable, so only they can be counted as equally likely

Two totals can occur at once, so they are not a valid sample space

The answer is: Only the pairs are interchangeable, so only they can be counted as equally likely

The pairs are easier to write down

The answer is: Only the pairs are interchangeable, so only they can be counted as equally likely

2.[2p]
Match each phrase to the set operation it becomes.

A or B
A and B
not A
A but not B

intersection
complement
union
difference

Show the answer

A or B: union A and B: intersection not A: complement A but not B: difference

3.[2p]
Which axiom is used to prove that $P (A^{c}) = 1 - P (A)$ ?

Additivity for disjoint events, applied to $A$ and $A^{c}$ , whose union is $Ω$

Correct

Non-negativity, applied to $A^{c}$

The answer is: Additivity for disjoint events, applied to $A$ and $A^{c}$ , whose union is $Ω$

Countable additivity, applied to an infinite sequence

The answer is: Additivity for disjoint events, applied to $A$ and $A^{c}$ , whose union is $Ω$

None: it is an axiom in its own right

The answer is: Additivity for disjoint events, applied to $A$ and $A^{c}$ , whose union is $Ω$

4.[2p]
$P (A) = 0.6$ , $P (B) = 0.5$ and $P (A \cap B) = 0.2$ . Find $P (A \cup B)$ .

CorrectNot quite: 0.9

5.[3p]
A whole number is drawn at random from $1$ to $1000$ . What is the probability that it is divisible by $2$ , by $3$ or by $5$ ?

CorrectNot quite: 0.734

6.[1p]
The union bound $P (A \cup B) \leq P (A) + P (B)$ requires $A$ and $B$ to be disjoint.

True

The answer is: False

False

Correct

7.[2p]
A die is loaded so that $P (6) = 0.25$ and the other five faces are equally likely. What is the probability of rolling an even number?

CorrectNot quite: 0.55

8.[3p]
An event has probability zero. What does that establish?

Nothing about whether it can occur: on an infinite sample space, possible events can have probability zero

Correct

That it is impossible

The answer is: Nothing about whether it can occur: on an infinite sample space, possible events can have probability zero

That its complement is impossible

The answer is: Nothing about whether it can occur: on an infinite sample space, possible events can have probability zero

That the sample space was chosen wrongly

The answer is: Nothing about whether it can occur: on an infinite sample space, possible events can have probability zero

9.[2p]
Put these results in the order the lesson derives them, each using the ones before it.

$P (A) \leq P (B)$ when $A \subseteq B$ , from splitting $B$ in two
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
$P (A_{1} \cup \dots \cup A_{n}) \leq P (A_{1}) + \dots + P (A_{n})$
$P (A^{c}) = 1 - P (A)$ , from additivity on $A$ and $A^{c}$

Show the answer

a, b, c, d

1.[2p] Why is the sample space for two dice usually taken as the 36 ordered pairs rather than the 11 possible totals?

2.[2p] Match each phrase to the set operation it becomes.

3.[2p] Which axiom is used to prove that P(Ac)=1−P(A)?

4.[2p] P(A)=0.6, P(B)=0.5 and P(A∩B)=0.2. Find P(A∪B).

5.[3p] A whole number is drawn at random from 1 to 1000. What is the probability that it is divisible by 2, by 3 or by 5?

6.[1p] The union bound P(A∪B)≤P(A)+P(B) requires A and B to be disjoint.

7.[2p] A die is loaded so that P(6)=0.25 and the other five faces are equally likely. What is the probability of rolling an even number?

8.[3p] An event has probability zero. What does that establish?

9.[2p] Put these results in the order the lesson derives them, each using the ones before it.

1.[2p]
Why is the sample space for two dice usually taken as the $36$ ordered pairs rather than the $11$ possible totals?

2.[2p]
Match each phrase to the set operation it becomes.

3.[2p]
Which axiom is used to prove that $P (A^{c}) = 1 - P (A)$ ?

4.[2p]
$P (A) = 0.6$ , $P (B) = 0.5$ and $P (A \cap B) = 0.2$ . Find $P (A \cup B)$ .

5.[3p]
A whole number is drawn at random from $1$ to $1000$ . What is the probability that it is divisible by $2$ , by $3$ or by $5$ ?

6.[1p]
The union bound $P (A \cup B) \leq P (A) + P (B)$ requires $A$ and $B$ to be disjoint.

7.[2p]
A die is loaded so that $P (6) = 0.25$ and the other five faces are equally likely. What is the probability of rolling an even number?

8.[3p]
An event has probability zero. What does that establish?

9.[2p]
Put these results in the order the lesson derives them, each using the ones before it.