CS580L Homework 0 Probability- Introduction to Machine Learning Solved

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All questions are worth 10 points. Total = 100 points.

Questions

If you have two standard six-sided dice, each with uniform probability of landing on each counting number from 1 to 6. What is the probability of rolling doubles (both dice landing on the same number)?
Let X and Y be two independent random variables. P[X,Y]= 0.2 and P[X] =0.5. Find P[Y].
A drunk person is walking on the road. With probability 0.6 he takes a step forward and with probability 0.4 he takes a step backward. After 10 steps, what is the probability that he is at his starting position? Just the expression is sufficient.
Let X, Y and Z be three random variables. E[X]= 2, Var(X) =1 and E[Y]=3. X and Y are independent of each other. Z = X2 Y. Find E[Z].
Find the mean, median and variance of the following numbers. 1, 6, -1, 4, 10.
A gambler bets n times. Each time the gambler bets, 20% of the time he wins $10 and 80% of the time he loses $5. What is expected gain (which can be negative) after n bets?
You are drawing cards from a deck (consisting of the standard 52 cards) one at a time without replacement. Let X and Y denote the first and second cards you draw from the deck. You observe that the first card is a spade. What is the probability that the second card you draw is also a spade?
Consider two urns. The first contains two white and seven black balls, and the second contains five white and six black balls. We flip a fair coin and then draw a ball from the first urn or the second urn depending on whether the outcome was heads or tails. What is the conditional probability that the outcome of the toss was heads given that a white ball was selected?
Suppose you toss a coin 10 times. The probability of getting a head in each toss is p. What is the probability that you get more than 6 heads in the 10 coin tosses. Just the expression is sufficient.
A man enters a betting competition. Each time the man bets, the probability of winning is p. What is the probability that he wins for the first time after n bets? Name this distribution.