Chapter 7: Discrete Probability

Question: Use Chebyshev's inequality to show that the probability that more than 10 people get the correct hat back when a hatcheck person returns hats at random does not exceed \(1/100\) no matter how many people check their hats.

(Hint: Use Example 6 and Exercise 43 in Section 7.4.)

Question: Let\(A\left( X \right) = E\left( {\left| {X - E\left( X \right)} \right|} \right)\), the expected value of the absolute value of the deviation of\(X\), where\(X\)is a random variable. Prove or disprove that\(A\left( {X + Y} \right) = A\left( X \right) + A\left( Y \right)\) for all random variables\(X\)and\(Y\).

Question: Suppose that 100 people enter a contest and that different winners are selected at random for first, second, and third prizes. What is the probability that Michelle wins one of these prizes if she is one of the contestants?

Question:Suppose that at least one of the events\({E_j},j = 1,2, \ldots ,m\), is guaranteed to occur and no more than two can occur. Show that if\(p\left( {{E_j}} \right) = q\)for\(j = 1,2, \ldots ,m\)and\(p\left( {{E_j} \cap {E_k}} \right) = r\)for\(1 \le j < k \le m\), then\(q \ge 1/m\)and\(r \le 2/m\).

Question: Suppose that 100 people enter a contest and that different winners are selected at random for first, second, and third prizes. What is the probability that Kumar, Janice, and Pedro each win a prize if each has entered the contest?

Question: Provide an example that shows that the variance of the sum of two random variables is not necessarily equal to the sum of their variances when the random variables are not independent

Question:Show that if\(m\)is a positive integer, then the probability that the\(m\)the success occurs on the\((m + n)\)the trial when independent Bernoulli trials, each with probability\(p\)of success, are run, is\(\left( {\begin{aligned}{{}{}}{n + m - 1}\\n\end{aligned}} \right){q^n}{p^m}\).

Question:Suppose that \({X_1}\) and \({X_2}\) are independent Bernoulli trials each with probability \(1/2\), and let \({X_3} = \left( {{X_1} + {X_2}} \right)\,\bmod \,2\).

a) Show that \({X_1}\),\({X_2}\) and \({X_3}\) are pairwise independent, but \({X_3}\) and \({X_1} + {X_2}\) are not independent.

b) Show that \(V\left( {{X_1} + {X_2} + {X_3}} \right) = V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + V\left( {{X_3}} \right)\).

c) Explain why a proof by mathematical induction of Theorem 7 does not work by considering the random variables \({X_1}\),\({X_2}\) and \({X_3}\).

Question: What is the probability that Abby, Barry and Sylvia win the first, second, and third prizes, respectively, in a drawing, if 200 people enter a contest and

(a) no one can win more than one prize.

(b) winning more than one prize is allowed.

Question: There are \(n\) different types of collectible cards you can get as prizes when you buy a particular product. Suppose that every time you buy this product it is equally likely that you get any type of these cards. Let \(X\) be the random variable equal to the number of products that need to be purchased to obtain at least one of each type of card and let \({X_j}\) be the random variable equal to the number of additional products that must be purchased after \(j\) different cards have been collected until a new card is obtained for \(j = 0,1, \ldots ,n - 1.\)

a) Show that\(X = \sum\limits_{j = 0}^{n - 1} {{X_j}} \).

b) Show that after\(j\)distinct types of cards have been obtained, the card obtained with the next purchase will be a card of a new type with probability\(\left( {n - j} \right){\rm{ }}/{\rm{ }}n.\)

c) Show that\({X_j}\)has a geometric distribution with parameter\(\left( {n - j} \right){\rm{ }}/{\rm{ }}n.\)

d) Use parts (a) and (c) to show that\(E(X) = n\sum\limits_{j = 1}^n 1 /j\).

e) Use the approximation\(\sum\limits_{j = 1}^n 1 /j \approx \ln n + \gamma \), where\(\gamma = 0.57721 \ldots \)is Euler's constant, to find the expected number of products that you need to buy to get one card of each type if there are 50 different types of cards.