91Ó°ÊÓ

Suppose a die is rolled twice. What are the possible values that the following random variables can take on? (i) The maximum value to appear in the two rolls. (ii) The minimum value to appear in the two rolls. (iii) The sum of the two rolls. (iv) The value of the first roll minus the value of the second roll.

Suppose a coin having probability \(0.7\) of coming up heads is tossed three times. Let \(X\) denote the number of h?ads that appear in the three tosses. Determine the probability mass function of \(X\). '

A ball is drawn from an urn containing three white and three black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes on indefinitely. What is the probability that of the first four balls drawn, exactly two are white?

An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?

A television store owner figures that 50 percent of the customers entering his store will purchase an ordinary television set, 20 percent will purchase a color television set, and 30 percent will just be browsing. If five customers enter his store on a certain day, what is the probability that two customers purchase color sets, one customer purchases an ordinary set, and two customers purchase nothing?

If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.

Suppose that we want to generate a random variable \(X\) that is equally likely to be either 0 or 1 , and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability \(p .\) Consider the following procedure: 1\. Flip the coin, and let \(0_{1}\), either heads or tails, be the result. 2\. Flip the coin again, and let \(0_{2}\) be the result. 3\. If \(0_{1}\) and \(0_{2}^{-}\) are the same, return to step 1 . 4\. If \(0_{2}\) is heads, set \(X=0\), otherwise set \(X=1\). (a) Show that the random variable \(X\) generated by this procedure is equally likely to be either 0 or 1 . (b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different, and then sets \(X=0\) if the final flip is a head, and sets \(X=1\) if it is a tail?

If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is \(\frac{1}{100}\), what is the (approximate) probability that you will win a prize (a) at least once, (b) exactly once, (c) at least twice?

A point is uniformly distributed within the disk of radius 1 . That is, its density is $$ \therefore \quad f(x, y)=C, \quad 0 \leqslant x^{2}+y^{2} \leqslant 1 $$ Find the probability that its distance from the origin is less than \(x, 0 \leqslant x \leqslant 1\).

Let \(X_{1}, X_{2}, \ldots ; X_{n}\) be independent random variables, each having a uniform distribution over \((0,1)\). Let \(M=\) maximum \(\left(X_{1}, X_{2}, \ldots, X_{n}\right) .\) Show that the distribution function of \(M, F_{M}(\cdot)\), is given by $$ F_{M}(x)=x^{n}, \quad 0 \leqslant x \leqslant 1 $$ What is the probability density function of \(M\) ?