Introduction to Mathematical Statistics
Probability questions.
Please refer to the .dox file
You can write down the answer in paper and submit. handwriting needs to be clear
Introduction to Mathematical Statistics
1. In the game Three Card Monte there are 3 cards, one of them a queen. The three cards are placed face down on a table. The dealer shuffles the cards quickly while they are face down on the table. The player selects a card; the goal is to find the queen.
A. What is the probability you DO NOT choose the queen with your initial guess?
B. Before showing you the card you have chosen, the dealer flips one of the cards. It is not the queen. The dealer then offers you the option to keep the card you have chosen or take the remaining card. If you decide to pick the remaining card, what is the probability you choose the queen?
C.
What is the probability you choose the queen with your initial guess?
1. I write each of the letters P, R, I, D, E on separate pieces of paper and put these into a hat. I close my eyes and begin picking the pieces of paper from the hat and reading off the letters written on the pieces of paper.
1. If I do NOT put the pieces of paper back into the hat after each choice and pick 5 pieces of paper what is the probability that the letters come out exactly in the order PRIDE?
1. If I put the pieces of paper back into the hat after each choice and pick 5 pieces of paper, then what is the probability that the letters come out exactly in the order PRIDE?
1. If I put the pieces of paper back into the hat after each choice and pick 5 pieces of paper, then what is the probability that the letters picked can be rearranged to spell PRIDE?
1. If I put the pieces of paper back into the hat after each choice and pick 5 pieces of paper, then what is the probability that there are exactly two P letters among the 5 letters picked?
1. If I put the pieces of paper back into the hat after each choice and continue to pick letters until the first P occurs, what is the probability the first P occurs as the 4th letter picked?
1. Suppose 1% of people have a certain genetic mutation. 90% of tests for the gene mutation detect it in those who have it while 9.6% of the tests falsely detect it in those who do not have it.
2. If a person is chosen at random, what is the probability the person tests positive for the genetic mutation?
2. If a randomly chosen person tests positive for the genetic mutation what is the probability they actually have it?
1. StatCo manufactures and sells desktop computers and monitors. Suppose that out of those customers who visit StatCo Computer Stores, 60% buy desktops, 60% buy monitors, and 90% buy either a monitor or a desktop or both.
1. What is the probability that a customer buys a desktop and a monitor?
1. What is the probability that a customer buys a desktop, given that they bought a monitor?
1. What is the probability that a customer buys a desktop, given that they did not buy a monitor?
1. A study was conducted to determine if a medicine helps to shorten the length of a cold. 200 subjects were recruited who had suffered from a cold and either took the medicine or not. The length of times in days the subject suffered from the cold was noted. The following table contains the results.
Medicine?
Cold Length
Yes
No
1-3 Days
86
19
4-7 Days
16
79
0. If a subject is chosen at random, what is the probability the subject both took the medicine and had the cold for 1-3 days?
0. If a subject is chosen at random, what is the probability the subject took the medicine given that the subjects cold lasted for 1-3 days?
1. Are the events subject took the medicine and cold lasted for 1-3 days independent or dependent? Why or why not?
