Applied-Probability-and-Statistics Exam Dumps : Applied Probability and Statistics (FZO1 C955)

“At least one six” means one six or two sixes across the two rolls. The most efficient method is to use the complement rule. The complement of at least one six is no six on either roll. On one die roll, the probability of not rolling a six is 5/6. Since the two die rolls are independent, the probability of no six on both rolls is (5/6)(5/6) = 25/36. Therefore, the probability of at least one six is 1 − 25/36 = 11/36. This includes all outcomes with a six on the first roll, a six on the second roll, or sixes on both rolls. Option B gives the probability of a six on one single roll only. Option C gives the probability of rolling two sixes. Option D gives the complement, no sixes. Study Guide references/topics: complement rule, independent events, dice probability, sample spaces.

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The probabilities of all outcomes in a complete sample space must sum to 1. A sample space contains every possible outcome of a probability experiment, and one of those outcomes must occur. For example, when rolling a fair six-sided die, the outcomes are 1, 2, 3, 4, 5, and 6. Each has probability 1/6, and the sum is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1. A total probability of 0 would mean no outcome can occur, which is impossible for a valid experiment. A total greater than 1 violates probability rules because probabilities cannot exceed certainty. “Cannot exceed 2” is too broad and mathematically invalid, since the exact total must equal 1. This principle is foundational for checking probability distributions and validating whether assigned probabilities are coherent. Study Guide references/topics: sample space, probability axioms, total probability, theoretical probability.

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For two events, the general addition rule is P(A or B) = P(A) + P(B) − P(A and B). Because A and B are independent, their joint probability is P(A and B) = P(A)P(B). Here, P(A) = 0.3 and P(B) = 0.2, so P(A and B) = 0.3 × 0.2 = 0.06. Then P(A or B) = 0.3 + 0.2 − 0.06 = 0.44. The subtraction is necessary because the overlap, where both A and B occur, is counted twice when adding P(A) and P(B). Option A, 0.5, incorrectly adds the two probabilities without correcting for overlap. Option B gives the probability of both events, not either event. Option D is unrelated to the addition rule. Study Guide references/topics: independent events, addition rule, multiplication rule, union of events.

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