Chapter 12: From Randomness to Probability

Question 1

Define ‘Random phenomenon’.

Answer 1

A phenomenon is random is we know what outcomes could happen, but not which particular values will happen.

Question 2

Define ‘Trial’.

Answer 2

A single attempt or realization of a random phenomenon’.

Question 3

Define ‘Outcome’.

Answer 3

The value measured, observed, or reported for an individual instance of a trial.

Question 4

Define ‘Event’.

Answer 4

A collection of outcomes. Usually, we identify events so that we can attach probabilities to them. We denote events with bold capital letters, such as A, B, or C.

Question 5

Define ‘Sample space’.

Answer 5

The collection of all possible outcome values. The sample space has a probability of 1.

Question 6

Define ‘Law of Large Numbers’.

Answer 6

The relative frequency of an event in repeated INDEPENDENT trials gets closer and closer to its true or long-run relative frequency as the number of trials increases. Note that it speaks only of long-run behavior and we need to be careful not to misinterpret. Even when we’ve observed a string of heads, we shouldn’t expect extra tails in subsequent coin flips.

Question 7

Define ‘Independence (informally)’.

Answer 7

Two events are independent if learning that one event occurs does not change the probability that the other event occurs.

Question 8

Define ‘Probability’.

Answer 8

The probability of an event is a number between 0 and 1 that reports the likelihood of an event’s occurrence. We write P(A) for the probability of the event A.

Question 9

Define ‘Empirical probability’.

Answer 9

The long-run relative frequency of an event’s occurrence. The Law of Large Numbers assures us of its existence, when we can perform repeated independent trials.

Question 10

Define ‘Theoretical probability’.

Answer 10

The probability that comes from a model (such as equally likely outcomes).

Question 11

Define ‘Personal probability’.

Answer 11

A probability that is subjective and represents your personal degree of belief.

Question 12

Define ‘Probability assignment rule’.

Answer 12

A probability is a number between 0 and 1. For any event A, 0 ≤ P(A) ≤ 1.

Question 13

Define ‘Total probability rule’.

Answer 13

Since the probability of any event is between 0 and 1, inclusive, the entire sample space must be 1. P(S) = 1.

Question 14

Define ‘Disjoint or mutually exclusive’.

Answer 14

Two events are disjoint if they share no outcomes in common. If A and B are disjoint, then knowing that A occurs tells us that B cannot occur. Disjoint events are also called mutually exclusive events.

Question 15

Define ‘Addition rule for disjoint events’.

Answer 15

If A and B are disjoint events, then the probability of A or B is P(A + B) = (P(A) + P(B).

Question 16

Define ‘Legitimate probability assignment’.

Answer 16

An assignment of probabilities to outcomes is legitimate if:

• each probability is between 0 and 1 (inclusive)
• the sum of the probabilities is 1.

Question 17

Define ‘Complement rule’.

Answer 17

The probability of an event occurring is 1 minus the probability that it doesn’t occur. P(A) = 1 - P(A^C).

Question 18

Define ‘General addition rule’.

Answer 18

For any two events, A and B, the probability of A or B is P(A or B) = P(A) + P(B) - P(A and B).

Question 19

Probability is usually based on ___-___ relative frequencies.

Answer 19

Long-run.

Question 20

What are some basic rules for combining probabilities of outcomes to find probabilities of more complex events?

Answer 20

• The probability assignment rule
• The total probability rule
• The additions rule for disjoint events
• The complement rule
• The general addition rule