C955 Applied Probability and Statistics - Set 3 - Part 1

Correct Answer: The overlap counted twice when events can both occur

When events can occur together, adding their probabilities counts the overlap twice, so the joint probability is subtracted to correct for the double counting.

Correct Answer: 0.7

Using the addition rule, 0.5 plus 0.4 minus 0.2 equals 0.7.

Correct Answer: P(A) times P(B given A)

The general multiplication rule accounts for dependence: the probability both occur equals the probability of A times the conditional probability of B given A.

Correct Answer: 3/10

The first red has probability 3/5. After removing it, the second red has probability 2/4. Multiplying gives 3/5 times 2/4 equals 6/20, which simplifies to 3/10.

Correct Answer: 120

5! equals 5 times 4 times 3 times 2 times 1, which is 120.

Correct Answer: Arrangements where order matters

A permutation counts ordered arrangements, so ABC and ACB are counted as different, unlike combinations where order is irrelevant.

Correct Answer: Selections where order does not matter

A combination counts the number of ways to choose a subset without regard to order, so ABC and ACB represent the same selection.

Correct Answer: 6

The number of arrangements of 3 distinct items is 3!, which equals 3 times 2 times 1, or 6.

Correct Answer: 6

The number of combinations of 4 items taken 2 at a time is 4! divided by (2! times 2!), which equals 24 divided by 4, giving 6.

Correct Answer: 20

Because the two roles are distinct, order matters. There are 5 choices for president and 4 remaining for vice president, giving 5 times 4 equals 20.

Correct Answer: The set of all possible outcomes

The sample space lists every possible outcome of an experiment, forming the basis for assigning probabilities to events.

Correct Answer: HH, HT, TH, TT

Each coin has two outcomes, and the coins are independent, so the four equally likely outcomes are HH, HT, TH, and TT.

Correct Answer: 36

Each die has 6 outcomes and the dice are independent, so the total number of ordered outcomes is 6 times 6, which equals 36.

Correct Answer: 6/36

The pairs summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), giving 6 favorable outcomes out of 36, which is 6/36 or 1/6.

Correct Answer: 7

A sum of 7 has the most combinations, six of them, more than any other total, making it the most probable outcome.

Correct Answer: A description of all possible values and their probabilities

A probability distribution specifies every possible value a random variable can take along with the probability of each, summing to 1 for discrete variables.

Correct Answer: All outcomes are equally likely

A uniform distribution assigns equal probability to every outcome, as with a single roll of a fair die where each face has probability 1/6.

Correct Answer: The spread of values around the expected value

Variance measures how much the values of a random variable deviate from the expected value, computed as the probability-weighted average of squared deviations.

Correct Answer: The number of events occurring in a fixed interval

The Poisson distribution models the count of independent events occurring at a constant average rate within a fixed interval of time or space.

Correct Answer: The number of calls a help desk receives per hour

Counting calls arriving at a roughly constant rate over an hour matches the Poisson model of independent events in a fixed interval.