301115- Advanced Statistical Methods Worksheet 1Solved

Probability
1.   Given the events A and B and their probability P(A) = 0.2, P(B) = 0.4 and P(A ∩ B) = 0.1, for each of the following compound events, draw the probability diagram and shade the region of interest and compute probability of the compound event.

•  P(AC)

•  P(A ∪ B)

•  P(A ∪ BC)

•  P((A ∩ B)C)

•  P(AC ∪ BC)

•  P(AC ∩ BC)

•  P((AC ∩ BC)C)

•  P((AC ∪ BC)C)

2.   A Magic the Gathering deck of cards contains 60 cards; 20 of the cards are land cards and the remaining 40 cards are non-land cards. If I draw seven cards from the deck at random, what is the probability that I get:

•  three land cards?

•  more than three land cards?

•  between two and four land cards?

What is the expected number of land cards when I draw seven cards at random (use Wikipedia to help with this)?

3.   You and a friend play a game where you shuffle a deck of 52 playing cards, then reveal the top card. If the card is of the suit “hearts”, then you win $2. If the top card is not of the suit “hearts”, then you give your friend $1 and you must play again.

•  What is probability that you reveal a “heart” after the first shuffle?

•  What is the probability that you don’t lose money?

•  What is the expected number of games that you must play and the expected winnings (use Wikipedia again)? 4. The birthday problem is a famous problem in probability, in which we calculate how many people we would need to gather in a room before there is at least a 0.5 probability that at least two of these people share a birthday. We assume that birthdays are uniformly distributed across the year, and that all birthdaysare independent (i.e. no twins). Use R to calculate the probability for a room with n people and plot the probabilities for various values of n. Use a reference line to show p = 0.5 on your plot. Choose sensible limits for the x and y axes.

Conditional Probability
1.   Given the events A and B and their probability P(A) = 0.2, P(B) = 0.4 and P(A ∩ B) = 0.1, for each of the following events, draw the probability diagram and shade the region of interest and compute probability of the events.

•  P(A|B)

•  P(B|A)

1

Are events A and B independent?

2.   The probability of living in the outback, given that you have seen a UFO is 0.4. The probability of living in the outback, given that you have not seen a UFO is 0.2. Given that the probability of seeing a UFO is 0.01, what is the probability of seeing a UFO, given that you live in the outback?

Random Variables
Given the following random variables, use Wikipedia to find the expected value and variance. Make sure to describe what the expected value means for each distribution.

•  X is Binomial, with number of trials n = 12 and probability of success p = 0.2

•  Y is Hypergeometric with number of trials n = 10, number of success states x = 25 and number of failure states

y = 45.

•  Z is Geometric with probability of success p = 0.1.

•  A is Normal, with mean µ = 10 and standard deviation σ = 5.

•  B is Poisson with rate λ = 4.5.

Bayes’ Theorem
Consider the following scenario. You have a disease that is present in 1% of the human population. You have a test for the disease that is 95% accurate, meaning that 95% of the time a person with the disease will receive a positive test result, and 5% of the time they will receive a negative test result. Conversely, a person without the disease will receive a positive test result 5% of the time, and a negative test result 95% of the time. Translate the above problem formulation into probability statements

(i.e. use D for the event that the individual has the disease and DC for the event that they do not, T for a positive test, TC for a negative test). Use Bayes rule to calculate the probability that a person with a positive test result has the disease (hint: you will need to apply the law of total probability to obtain P(T)). Does the result surprise you? Use R to plot the accuracy of the

test against the probability that a person with a positive test result has the disease for accuracies of between 0% and 100%.

2