# [Solved] 301115 -Advanced Statistical Methods-Worksheet 1

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# 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)?

1. 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

# 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?

1. 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%.

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[Solved] 301115 -Advanced Statistical Methods-Worksheet 1
30 \$