Probability Questions and Answers

Answer: A

Probabilities run from 0 (impossible) to 1 (certain), so the less likely an event, the closer its probability sits to 0.

\[ 0.0001 < 0.001 < 0.01 < 0.1 \]

The smallest value, 0.0001, describes the least likely event - about one chance in ten thousand.

Answer: C

A random experiment is one whose set of possible outcomes is known in advance but whose particular outcome on any trial cannot be predicted - rolling a die, for instance.

The other terms name different things: the sample space is the set of outcomes, and an elementary event is a single one of them. Neither is the experiment itself.

Answer: D

A deterministic experiment gives the same result every time it is repeated under identical conditions - heating water to 100°C at sea level, for example.

A random experiment is the opposite: the conditions may be identical, yet the outcome varies from trial to trial. Probability theory is concerned with the second kind.

Answer: B

An event is any subset of the sample space - one outcome or a collection of them.

Rolling an even number, for example, is the event \( \{2, 4, 6\} \).

Probability is the number assigned to an event, and a random variable is a function from outcomes to numbers - neither is the collection itself.

Answer: C

A standard deck has four fives and four sevens, and no card is both.

\[ P = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13} \]

Because the two events are mutually exclusive, the probabilities simply add with no overlap to subtract.

Answer: A

The special rule of multiplication states that

\[ P(A \cap B) = P(A) \times P(B) \]

and this holds only when the events are independent - the occurrence of one leaving the probability of the other unchanged.

Mutual exclusivity belongs to the special rule of addition, \( P(A \cup B) = P(A) + P(B) \). The two conditions are quite different: mutually exclusive events with non-zero probabilities are in fact never independent, since one occurring makes the other impossible.

Answer: D

A probability is a number attached to an event, not a collection of experiments.

The three true statements are that it lies between 0 and 1, that it may be written as a decimal or a fraction, and that a value near 0 marks an unlikely event.

The false one is that it is the collection of several experiments - that describes a sequence of trials, not the probability itself.

Answer: A

A deck has 13 diamonds and 13 clubs, and no card belongs to both suits.

\[ P = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} \]

The two suits are mutually exclusive, so no overlap is subtracted. Diamonds and clubs together make up exactly half the pack.

Answer: A

A discrete distribution takes values that can be listed - whole counts or categories.

• Number of bedrooms - 1, 2, 3, … discrete
• Number of bathrooms - discrete
• Whether there is a pool - two categories, discrete
• Sale price - any value in a range, continuous

So the sale or purchase price of a house is the odd one out: between any two prices there is always another possible price.

Answer: D

Let \( A \) be "at least one die shows 4" and \( B \) be "the sum is 7", over the 36 outcomes.

\[ |A| = 11 \quad (\text{six with the first a 4, six with the second, less the double-counted } (4,4)) \]

\[ |B| = 6: \ (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) \]

\[ |A \cap B| = 2: \ (3,4),(4,3) \]

\[ |A \cup B| = 11 + 6 - 2 = 15 \quad\Longrightarrow\quad P = \frac{15}{36} \]

Both corrections matter: \( (4,4) \) within \( A \), and the two outcomes shared by \( A \) and \( B \).