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Chapter 1: Problem 189

A gambling game is played as follows: \(D\) dollar bills are distributed in some manner among \(N\) indistinguishable envelopes, which are then mixed up in a large bag. The player buys random envelopes, one at a time, for one dollar and examines their contents as they are purchased. If the player can buy as many or as few envelopes as desired, and, furthermore, knows the initial distribution of the money, then for what distribution(s) will the player's expected net return be maximized?

Short Answer

Expert verified
Distribute the dollar bills evenly among the envelopes.

Step by step solution

01

Define Variables

Let the total amount of money be denoted as \(D\) and the number of envelopes be denoted as \(N\). Each envelope has an amount of money represented by a random variable \(X_i\) where \(i = 1, 2, \text{...}, N\).
02

Calculate Expected Return

To determine the expected net return, consider that the player buys envelopes one by one for \(1 each. The expected return from opening an envelope can be represented by the expected value of \)X_i\( minus the cost of buying the envelope (which is \)1).
03

Determine the Expected Value of \(X_i\)

The expected value \(E(X_i)\) is calculated as the total amount of money \(D\) divided by the number of envelopes \(N\), that is, E(X_i) = \frac{D}{N}.
04

Calculate the Expected Net Return

The expected net return from each envelope is given by the formula \(E(X_i) - 1\). Substituting in the expression for \(E(X_i)\) gives E(X_i) - 1 = \frac{D}{N} - 1.
05

Maximize Expected Net Return

For the player's expected net return to be maximized, \(\frac{D}{N} - 1\) should be maximized. This implies that the expected value \(\frac{D}{N}\) should be greater than \(1\). The best strategy is to distribute \(D\) dollars as evenly as possible among \(N\) envelopes so that \(X_i\) is close to \(\frac{D}{N}\) for all \(i\).
06

Optimal Distribution Strategy

The optimal distribution occurs when \(D\) is evenly divisible by \(N\) and each envelope contains exactly \(\frac{D}{N}\) dollars. This ensures that every envelope has an equal amount, maximizing the expected net return.
07

Conclusion

Therefore, to maximize the player's expected net return, distribute the dollar bills as evenly as possible among the envelopes so that each envelope contains \(\frac{D}{N}\) dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Distribution
A probability distribution describes how probabilities are assigned to values of a random variable. In the gambling game discussed, the distribution of dollar bills across the envelopes creates a particular probability distribution. This is because each envelope can be seen as a random variable with a certain value (amount of money), and the way we distribute the dollars (evenly or unevenly) impacts the overall probability distribution of those values. The goal in this game is to determine a distribution that will maximize the player's expected net return.
Expected Value
The expected value is a key concept in probability theory that provides a measure of the center of a probability distribution. In the context of the gambling game, it represents the average amount of money you can expect to find in any envelope.
Mathematically, the expected value of money in an envelope, represented as \( E(X_i) \), is given by the total amount of money divided by the number of envelopes, \( E(X_i) = \frac{D}{N} \).
The expected value helps determine the expected net return, calculated as \( E(X_i) - 1 \), where 1 is the cost of purchasing one envelope. By maximizing this expected value, you can increase your net returns.
Optimization Strategies
Optimization strategies aim to find the best possible distribution of dollar bills that will maximize the player's expected net return. From the solution, we see that the strategy that works best is an equal distribution of money among envelopes. This means that if \( D \) dollars are distributed evenly among \( N \) envelopes, each envelope contains \( \frac{D}{N} \) dollars.
By following this strategy, we ensure the expected value of each envelope is maximized, and hence the net return is also maximized. This uniform distribution avoids the risk associated with high variability in the amounts within each envelope.
Random Variables
In probability theory, a random variable is a variable whose value is subject to variations due to randomness. In our gambling game, \( X_i \) is a random variable that represents the amount of money in the \( i \)-th envelope. Each \( X_i \) can take on different values depending on the distribution of the money.
The sum of all these random variables \( \sum_{i=1}^{N} X_i \) will always be equal to \( D \), the total amount of money. Understanding the behavior of these random variables helps in predicting overall outcomes and making informed decisions about which strategy to employ for maximizing returns.
Probability Theory
Probability theory is the branch of mathematics that deals with the analysis of random phenomena. It provides the foundation for understanding and modeling the gambling game. By applying the principles of probability theory, we are able to calculate the expected value and optimize strategies.
It helps us predict the likelihood of different outcomes and make sense of how different distributions of money will affect the player鈥檚 net returns. Probability theory thus guides us in formulating strategies that maximize the expected net return, such as the even distribution strategy highlighted in the solution.

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Most popular questions from this chapter

Note that if we tile the plane with black and white squares in a regular "checkerboard" pattern, then every square has an equal number of black and of white neighbors (four each), where two squares are considered neighbors if they are not the same but they have at least one common point. If we try the analogous pattern of cubes in 3-space, it no longer works this way: every white cube has 14 black neighbors and only 12 white neighbors, and vice versa. a. Show that there is a different color pattern of black and white "grid" cubes in 3-space for which every cube does have exactly 13 neighbors of each color. b. What happens in \(n\)-space for \(n>3\) ? Is it still possible to find a color pattern for a regular grid of "hypercubes" so that every hypercube, whether black or white, has an equal number of black and white neighbors? If so, show why; if not, give an example of a specific \(n\) for which it is impossible.

Define a function \(f\) by \(f(x)=x^{1 / x^{x^{1 / x^{x^{1 / x ^{. ^{. ^{. }}}}}}}} (x>0)\). That is to say, for a fixed \(x\), let $$ a_1=x, \quad a_2=x^{1 / x}, \quad a_3=x^{1 / x^x}=x^{1 / x^{a_1}}, \quad a_4=x^{1 / x^{x^{1 / x}}}=x^{1 / x^{a_2}}, \quad \ldots $$ and, in general, \(a_{n+2}=x^{1 / x^{a_n}}\), and take \(f(x)=\lim _{n \rightarrow \infty} a_n\). a. Assuming that this limit exists, let \(M\) be the maximum value of \(f\) as \(x\) ranges over all positive real numbers. Evaluate \(M^M\). b. Prove that \(f(x)\) is well defined; that is, that the limit exists.

Let \(g\) be a continuous function defined on the positive real numbers. Define a sequence \(\left(f_n\right)\) of functions as follows. Let \(f_0(x)=1\), and for \(n \geq 0\) and \(x>0\), let $$ f_{n+1}(x)=\int_1^x f_n(t) g(t) d t $$ Suppose that for all \(x>0, \sum_{n=0}^{\infty} f_n(x)=x\). Find the function \(g\). \(\quad\)

It's not hard to see that in the plane, the largest number of nonzero vectors that can be chosen so that any two of the vectors make the same nonzero angle with each other is 3 (and the only possible nonzero angle for three such vectors to make is \(2 \pi / 3)\). Now suppose we have vectors in \(n\) dimensional space. What is the largest possible number of nonzero vectors in \(n\)-space so that the angle between any two of the vectors is the same (and not zero)? In that situation, what are the possible values for the angle?

Let \(A\) be a set of \(n\) real numbers. Because \(A\) has \(2^n\) subsets, we can get \(2^n\) sums by choosing a subset \(B\) of \(A\) and taking the sum of the numbers in \(B\). (By convention, if \(B\) is the empty set, that sum is 0.) What is the least number of different sums we must get (as a function of \(n\) ) by taking the \(2^n\) possible sums of subsets of a set with \(n\) numbers?
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