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 ≤ 8 occurs, i.e., P(x1 + x2 ≤ 8 | x1 + x2 + x3 = 17 and x1, x2, x3 ∈ Z+) (Z+ is the 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%)

Using the following data (already sorted), use a goodness of fit test to test whether it comes from an exponential distribution. The exponential distribution has one parameter, its mean, μ (which is also its standard deviation). The exponential distribution is a continuous distribution that takes on only positive values in the interval (0,∞). Probabilities for the exponential distribution can be found based on the following probability expression:

.

Use 10 equally likely cells for your goodness of fit test.

Data Display

0.2 0.4 0.5 0.5 0.7 0.8 1.0 1.2 1.2 1.2

1.4 1.5 1.5 1.6 1.7 1.7 1.7 1.8 1.8 1.9

2.0 2.3 2.6 2.7 2.7 2.8 2.8 2.8 2.8 2.8

2.8 2.9 2.9 3.0 3.0 3.0 3.2 3.2 3.2 3.4

3.4 3.5 3.6 3.6 3.7 3.8 3.9 3.9 3.9 4.0

4.1 4.1 4.2 4.3 4.5 4.5 4.5 4.6 4.7 4.8

4.8 4.9 4.9 4.9 5.0 5.0 5.1 5.1 5.1 5.3

5.3 5.3 5.3 5.4 5.4 5.4 5.4 5.5 5.5 5.5

5.6 5.6 5.6 5.7 5.7 5.8 5.8 5.8 5.9 5.9

6.0 6.0 6.2 6.2 6.2 6.3 6.3 6.3 6.3 6.4

6.6 6.6 6.6 6.6 6.7 6.8 6.9 6.9 6.9 7.0

7.0 7.1 7.2 7.3 7.3 7.4 7.5 7.5 7.6 7.6

7.7 7.8 7.8 7.9 8.0 8.0 8.0 8.1 8.1 8.1

8.2 8.3 8.4 8.4 8.4 8.5 8.5 8.6 8.6 8.7

8.7 8.8 9.0 9.1 9.1 9.2 9.3 9.4 9.5 9.6

9.6 9.6 9.8 9.9 9.9 9.9 10.0 10.1 10.2 10.5

10.6 10.7 10.7 10.8 10.9 10.9 11.0 11.0 11.4 11.5

11.7 11.8 11.8 11.9 12.0 12.0 12.1 12.1 12.3 12.3

12.3 12.3 12.6 12.9 13.1 13.3 13.3 13.4 13.5 13.6

13.9 14.0 14.2 14.2 14.3 14.3 14.4 15.0 15.0 15.2

15.6 15.6 15.7 15.7 15.7 15.9 16.0 16.3 16.4 16.5

16.5 16.6 16.6 16.7 17.2 17.3 17.3 17.4 17.7 17.9

18.6 18.8 19.9 19.9 19.9 20.0 20.1 20.3 20.4 21.0

21.3 21.5 22.2 23.3 23.5 23.9 24.3 24.8 25.5 25.5

25.6 25.8 27.5 28.2 30.9 35.7 36.3 37.2 40.9 52.8

Descriptive Statistics:

Variable    N   Mean

Exp?      250 9.974

A bank wants to know if the enrollment on their website is above 30%. They test the null hypothesis that the proportion is 0.3, and the alternative that the proportion is greater than 0.3. After obtaining a sample, they fail to reject the null hypothesis. Later they find out that 30% of all customers enrolled. Answer whether there was a Type I error, Type II error, or neither error was made.

Group of answer choices

Type I error

Type II error

Neither error was made

Assume Raoult's Law applies to a binary mixture of water and methanol at 70 C.

a) Which species should be considered species (a)? Why?

b) Prepare a Pxy diagram for this mixture

c) What is the composition of the vapor in equilibrium with liquid at x_i for methanol of 0.3?

d) What is the composition of the liquid in equilibrium with vapor at y_i for water of 0.3?

For parts (c) and (d), determine these composition values analytically. Then confirm that they are correct on the Pxy diagram you created in part (b).

four squirrels were found to have an average weight of 9.0 ounces with a sample standard deviation is 0.7. Find the 95% confidence interval of the true mean weight.

Let x be a binomial random variable with n=7 and p=0.7. Find the following.

P(X = 4)

P(X < 5)

P(X ≥ 4)