Question

2. Probability (30%). Figure out the probability in the following scenarios.

(a) A number generator is able to generate an integer in the range of [1, 100], where each number has equal chances to be generated. What is the probability that a randomly generated number x is divisible by either 2 or 3, i.e., P(2 | x or 3 | x)? (5%)

(b) In a course exam, there are 10 single-choice questions, each worthing 10 points and having 4 choices (A, B, C, and D, with only one correct). There is one student, denoted as s, who has learned nothing from the course and hence has to randomly guess the answers. That means for any question, each one of the four choices has equal chances to be picked up by s. What is the probability that s passes the exam (earning a total of ≥ 60 points)? (5%)

(c) Consider three positive integers, x1, x2, x3, which satisfy the inequality below:
x1 +x2 +x3 =17. (1)

Let’s assume each element in the sample space (consisting of solution vectors (x1, x2, x3) satisfying the above conditions) is equally likely to occur. For example, we have equal chances to have (x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the probability the events x1 +x2 ≤8occurs,i.e.,P(x1 +x2 ≤8|x1 +x2 +x3 =17andx1,x2,x3 ∈Z+)(Z+ isthe set containing all the possible positive integers)? (5%)

(d) There are unlimited fake coins and only one real coin. The fake coins and the real coin are almost the same and can only be detected by a special machine. At the very beginning, there are two coins in a bag, one fake and the other real (but we don’t know which one is real). We continue the following process till the real coin is found: At the each step, we randomly sample one coin from the bag and examine whether it is fake. If yes, we put the coin back to the bag, additionally put in another fake coin, and randomly draw a coin for examination. The sampling process won’t stop until we find the real coin. Assuming that each coin (either fake or real) has equal chances to be selected, what is the probability that we sample 9 times but still cannot find the real coin (and hence has to continue the sampling process)? (5%)

(e) From a random sports news, the probability of observing the word “ball” and “player” is 0.8 and 0.7, respectively. For a non-sports news, the probability to observe “ball” is 0.1, so does that to observe “player”. Let’s assume that in any article, the appearance of any two words (including “ball” and “player”) are independent with each other. Also, the probability of sports news’ occurrences is 0.2. Given a news report x containing both “ball” and “player”, what is the probability that x is a sports news. (10%)

Answer 1

Thus the required probability is



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