Statistics and probability with R 1. A box contains 20 consecutive balls numbered 1 to 20. If four numbers are drawn at random, how many ways are there for the largest number to be a 13 and the smallest number to be 3 ? Use the Fundamential Principle of Counting! **Solution:** YOUR CODE HERE: 2. On a multiple-choice exam with **four** possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing? Hint: Use the fact that $P(E F) = P(E) P(F)$ for two independant event (generalized for more than events) Getting one answer correct is independant of another. Also $$P(at least 4) = P(exactly 4) + P(exactly 5)$$, the last events are mutually exclusive so $P(A \cup B) = P(A) + P(B)$ YOUR CODE HERE: Out of 200 students who addopt this test taking approach how many are expected to get at least 4 correct? **Hint:** Use Binomial experiment settings to answer this questions. Solution: 3. Consider tossing **four** fair coins. There are 16 possible outcomes, e.g `HHHH,HHHT,HHTH,HTHH,THHH, ...` possible outcomes. Define `X` as the random variable "number of heads showing when four coins are tossed." Obtain the mean and the variance of X. Simulate tossing three fair coins 10,000 times. Compute the simulated mean and variance of X. Are the simulated values within 2% of the theoretical answers? Hint: to find the theoretical values use `dbinom (x= , size = , prob =)` **Solution:** YOUR CODE HERE: ```{r} library(MASS) ```