Problem 16 - Applied Probability

Question

A sequence of trials (not necessarily independent) is performed. Let \(E_{i}\) be the event of success on the ith component trial. We associate with each trial a "payoff function" \(X_{i}=a I_{E_{i}}+b I_{E_{i}^{c}}\). Thus, an amount a is earned if there is a success on the trial and an amount \(b\) (usually negative) if there is a failure. Let \(S_{n}\) be the number of successes in the \(n\) trials and \(W\) be the net payoff. Show that \(W=(a-b) S_{n}+b n\)

Step-by-Step Solution

Answer

The net payoff is given by \(W = (a-b)S_n + bn\), where \((a-b)\) is the incremental gain from each success and \(bn\) accounts for the base payoff from all trials, adjusted for the total number of successes \(S_n\).

Step 1: Define the Payoff for Each Trial

The payoff for each trial is defined by the function \(X_i\), which depends on the success of the trial. If the event of success \(E_i\) occurs, the payoff is \(a\). If the success does not occur and the complementary event \(E_i^c\) happens instead, the payoff is \(b\). This can be written as \(X_i = aI_{E_i} + bI_{E_i^c}\), where \(I_{E_i}\) and \(I_{E_i^c}\) are indicator functions that are 1 if the corresponding event occurs and 0 otherwise.

Step 2: Express the Net Payoff W

The net payoff \(W\) is the sum of the payoffs of all \(n\) trials. We sum up the individual payoffs \(X_i\) to obtain \(W = \sum_{i=1}^{n} X_i = \sum_{i=1}^{n} (aI_{E_i} + bI_{E_i^c})\).

Step 3: Simplify the Expression for W

We can break the sum into two parts: one for successes and one for failures. The sum of the indicator function for successes is the total number of successes \(S_n\). For failures, since \(I_{E_i}\) and \(I_{E_i^c}\) are complementary (one is 1 and the other is 0 for a given trial), for \(n\) trials we will have \(n - S_n\) failures. This simplifies the expression for \(W\) to \(W = aS_n + b(n - S_n)\).

Step 4: Conclude the Final Expression for W

Rearrange the terms in the expression from Step 3 to group the \(S_n\) and \(n\) terms. This results in \(W = aS_n + bn - bS_n\). Factor out \(S_n\) to get \(W = S_n(a - b) + bn\). This is the final expression for the net payoff.

Key Concepts

Indicator Function

Understanding the concept of an indicator function is crucial when dealing with probabilistic models and random processes. In layman's terms, an indicator function is like a binary switch: it can be either 0 or 1. It is used to note the occurrence or non-occurrence of an event. For example, if we have an event called 'Success', denoted by \(E_i\), the indicator function \(I_{E_i}\) will be 1 when the event occurs and 0 when it does not. Similarly, if the complementary event \(E_i^c\), which denotes failure, occurs, then \(I_{E_i^c}\) will be 1.

In the context of the provided exercise, these indicator functions are integral to computing the payoff for each trial. If a trial is successful, the payoff function has a term involving \(I_{E_i}\) and a corresponding value 'a'. If it's a failure, the payoff function includes \(I_{E_i^c}\) with a value 'b'. The beauty of using indicator functions in this manner is that they simplify the process of calculating cumulative outcomes over multiple trials.

Net Payoff Calculation

Net payoff calculation is a fundamental operation when it comes to understanding economic, financial, or gambling scenarios. It represents the total gains after accounting for both wins and losses over a series of events. In our example, it involves calculating the overall payoff after a sequence of trials where each outcome might have different monetary consequences.

The step-by-step method outlined in the problem introduces the net payoff \(W\) as a sum of individual trial payoffs. Initially, it may seem complex due to the two different amounts earned based on the trial's result—success or failure. However, by isolating the number of successes \(S_n\) and recognizing the total number of trials \(n\), we can reformulate the payoff calculation in a more straightforward manner. The final expression \(W=(a-b)S_n + bn\) neatly encapsulates the net result, consolidating the individual trial results into an easily understandable outcome that reflects the total balance between successes and failures over all the trials.

Sequence of Trials

A sequence of trials represents a series of events or experiments where each trial has two possible outcomes—typically categorized as success or failure. In probabilistic terms, this is akin to flipping a coin repeatedly, where each flip is independent of the others. However, it is important to note that in our exercise, the trials are not necessarily independent.

The overall outcome after a sequence of trials can be non-intuitive because it involves accumulating the results of each individual trial. In the educational exercise we're exploring, we pay special attention to the number of successful trials \(S_n\) out of the total number of trials \(n\). Successes and failures contribute differently to the net payoff, which underscores the essence of understanding sequences of trials in contexts like risk assessment, strategic decision-making, and the analysis of random processes.