Question: What is the conditional probability that exactly 2 heads appear in 4 independent tosses of a fair coin, given that the first toss lands tails?

What is the conditional probability that exactly 2 heads appear in 4 independent tosses of a fair coin, given that the first toss lands tails?

Suppose we have a damaged die in which the 6 shows like a 3. Assume we throw the die twice, the throws being independent, and record values A and B. 1. What is the expected value of A + B? 2. What is the expected value of the product AB? 3. What is the expected value of B given A = 2?

Suppose you bought an intelligent AI machine to solve the homework problems. However, in all machines shipped from the manufacturer, 99% of them are good and 1% of them are bad. The good machines have probability 1/30 of failing after solving a problem. The bad machines have probability 1/2 of failing after solving a problem. Let T denote the number of problems solved until the machine fails, what is E[T]?

100 homeworks are on the table, with two questions to be graded on each homework. Aditya is in charge of grading question one and Jingjing is in charge of grading question two. First, Aditya grades some homeworks at random; each homework has probability 0.3 of being graded. Next, Jingjing randomly grades half of the homeworks. That is, he grades 50 homeworks. Assume Aditya and Jingjing make their choices independently 1. Let N be the number of homeworks that Aditya graded. What is E[N]? 2. Let M be the number of homeworks that Aditya graded and Jingjing did not grade. What is E[M]?

A 52-card deck is thoroughly shuffled and you are dealt a hand of 13 cards. 1. If you have one ace, what is the probability that you have a second ace? 2. If you have the ace of spades, what is the probability that you have a second ace?

A random variable X can have the value -1, 1, 2, or 1/2. Suppose that P(X = 2) = P(X =1/2)=1/3. Find values a = P(X = 1) and b = P(X = -1) such that E(X) = 1.

Urn A contains 1 red and 3 blue balls, urn B contains 2 red and 5 blue balls. One randomly chosen ball is transferred from urn B to urn A. Next, a random ball is chosen from urn A. What is the probability that this ball is red?

A deck of 52 cards is in random order, and the cards are turned up, one at a time. (a) What is the probability that the fifth card turned up is an ace? (b) What is the probability that the fifth card turned up is the first ace? (c) Given that the fifth card turned up is the first ace, what is the probability that the next card is the ace of spades?

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