C955 Applied Probability and Statistics - Set 3 - Part 3

Correct Answer: 1 minus the probability of no 6 in either roll

Computing the complement is simpler: the probability of no 6 in a roll is 5/6, so no 6 in two rolls is (5/6) squared, and 1 minus that gives at least one 6.

Correct Answer: 25/36

Each roll avoids a 6 with probability 5/6, and the rolls are independent, so the probability of no 6 in both is 5/6 times 5/6, which equals 25/36.

Correct Answer: 11/36

Using the complement, 1 minus 25/36 equals 11/36, the probability of getting at least one 6.

Correct Answer: Odds express the ratio of favorable to unfavorable outcomes

Odds compare favorable outcomes to unfavorable ones, so a probability of 1/4 corresponds to odds of 1 to 3, whereas probability compares favorable outcomes to all outcomes.

Correct Answer: 1 to 4

A probability of 0.2 means 1 favorable outcome out of 5, so 1 favorable and 4 unfavorable, giving odds of 1 to 4.

Correct Answer: Comparing the long-run average outcome of different choices

Expected value summarizes the long-run average result of a choice, allowing rational comparison of options under uncertainty, though it does not predict any single result.

Correct Answer: The potential loss may be unacceptable despite favorable average odds

Expected value ignores risk tolerance. A bet with positive average value but a chance of catastrophic loss may reasonably be declined, which motivates the concept of utility.

Correct Answer: Theoretical is calculated from the model; experimental is observed from trials

Theoretical probability is derived from the assumed model, while experimental (empirical) probability is estimated from observed relative frequencies, converging toward the theoretical value over many trials.

Correct Answer: It tends to approach the theoretical probability

By the law of large numbers, the experimental probability estimated from observed frequencies converges toward the true theoretical probability as trials accumulate.

Correct Answer: If one task has m outcomes and another has n, together they have m times n outcomes

The multiplication principle states that if one choice can be made in m ways and an independent second choice in n ways, the number of combined outcomes is m times n.