Problem 43 - Discrete Mathematics with Applications

Question

Let \(p\) denote the probability of success in a Bernoulli trial. Prove that the expected number of successes in a sequence of \(n\) Bernoulli trials is \(n p .\) (Hint: Use the binomial theorem.)

Step-by-Step Solution

Answer

In a single Bernoulli trial, the probability distribution function is given by \(P(X=k)\), where \(X\) denotes the outcome (success or failure) and \(k \in \{0,1\}\). The expectation of a single trial is \(E[X] = p\). For a sequence of \(n\) Bernoulli trials, the expected number of successes is the sum of the expected number of successes in each trial. Since each trial is independent, we can compute the expectation of the total number of successes by multiplying the expected number of successes in a single trial by the total number of trials, resulting in \(E[\text{Total successes}] = n E[X] = n p\). Therefore, the expected number of successes in a sequence of \(n\) Bernoulli trials is \(np\).

Step 1: Define the Bernoulli trial distribution and expectation

A Bernoulli trial is a random experiment where the outcome can either be success (denoted by \(1\)) or failure (denoted by \(0\)). In this exercise, the probability of success is given by \(p\), so the probability of failure is \(1-p\). To compute the expected number of successes in \(n\) trials, we first need to understand the distribution of the outcome in a single trial.

Step 2: Probability distribution function of the Bernoulli trial

The probability distribution function of a Bernoulli trial is given by: \(P(X=k) = \begin{cases} p, & \text{for}\ k=1 \\ 1-p, & \text{for}\ k=0 \end{cases}\) where \(X\) denotes the random variable representing the outcome (success or failure) and \(k \in \{0,1\}\).

Step 3: Definition of expectation

The expectation of a random variable \(X\) for a discrete probability distribution is defined as: \(E[X] = \sum_{k} k P(X=k)\) where the sum is taken over all possible values of \(k\).

Step 4: Calculate the expectation of a single Bernoulli trial

Now, we will calculate the expectation of a single Bernoulli trial using the distribution function from Step 2 and the definition of expectation from Step 3. \(E[X] = \sum_{k=0}^{1} k P(X=k) = 0(1-p) + 1(p) = p\) So, the expected number of successes in a single Bernoulli trial is \(p\).

Step 5: Calculate the expectation for \(n\) Bernoulli trials

In a sequence of \(n\) Bernoulli trials, the expected number of successes is the sum of the expected number of successes in each trial. Since each trial is independent, we can compute the expectation of the total number of successes simply by multiplying the expected number of successes in a single trial by the total number of trials, i.e., \(E[\text{Total successes}] = n E[X] = n p\) Where \(E[X] = p\) is the expected number of successes in a single trial, and \(n\) is the number of trials.

Step 6: Conclusion

Through the use of the binomial theorem and the definition of expectation, we have proven that the expected number of successes in a sequence of \(n\) Bernoulli trials is \(np\).

Key Concepts

Probability of Success

In a Bernoulli trial, the probability of success represents the likelihood of achieving a desired outcome in an experiment that has only two possible results: success or failure. This probability is denoted by \( p \), where \( 0 \leq p \leq 1 \).

Understanding the concept of probability is essential in determining outcomes in any given scenario. In each individual Bernoulli trial:

• Succcess is assigned a value of 1.
• Failure is assigned a value of 0.

The complementary probability—that is, the probability of failure—is simply \( 1 - p \). This forms the foundation of analyzing success or failure over multiple trials.

Expected Value

The expected value in probability theory reflects the average outcome expected over numerous trials. It's calculated as a weighted average of all possible values. For discrete random variables, the expected value is found using the formula:
\[E[X] = \sum_{i} x_i P(X=x_i)\]
Where:

• \(x_i\) is the value of each outcome.
• \(P(X = x_i)\) is the probability of each outcome occurring.

When dealing with a single Bernoulli trial, the expected value comes out to be the probability of success itself, with the formula:

\( E[X] = 0 \times (1-p) + 1 \times p = p \).
For \( n \) trials, the expected value becomes the sum of expected values for each trial, yielding \( np \), which translates to the average number of successes expected per sequence of trials.

Random Variables

Random variables are essential components in probability theory and statistics. They are variables that can take on widely different outcomes. A random variable, often denoted by \(X\), does not represent a fixed number but rather a range of potential outcomes based on probabilities. In a Bernoulli trial:

• A success is described with \(X = 1\).
• A failure with \(X = 0\).

This binary nature of outcomes makes Bernoulli trials and their sequences key cases in studying random variables.
Random variables help us understand and compute probability distributions, measure variances, and predict outcomes, essential when strategizing in uncertain environments, be it in science, finance, or day-to-day decision-making.

Binomial Theorem

The Binomial Theorem plays a crucial role in probability theory, especially while dealing with discrete outcomes like successes and failures. In essence, it describes the algebraic expansion of powers of a binomial, \((a + b)^n\). For Bernoulli trials, consider:
\[(p + (1-p))^n\]
The coefficients of this expansion give rise to probabilities in binomial distribution.
When calculating the probability of achieving exactly \(k\) successes in \(n\) trials, the theorem provides a formula:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
This formula encapsulates the probability of a random variable (here, the number of successes) being exactly \(k\), factoring in the inherent nature of Bernoulli trials as independent events.
Thus, the Binomial Theorem intertwines algebraic expressions with probability theory, being a cornerstone in calculating expected numbers and variances across trials.