Problem 11 - A First Course in Probability

Question

Consider \(n\) independent flips of a coin having probability \(p\) of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if \(n=\) 5 and the outcome is \(H H T H T,\) then there are 3 changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of \(n-1\) Bernoulli random variables.

Step-by-Step Solution

Answer

The expected number of changeovers in \(n\) independent flips of a coin with probability \(p\) of landing head is \(E(Y)=(n-1)2p(1-p)\).

Step 1: Define the Bernoulli Random Variables

Let's define the Bernoulli random variables \(X_1, X_2,\ldots,X_{n-1}\), where \(X_i\) equals 1 if a changeover occurs between the i-th and (i+1)-th flip, and 0 otherwise. So for each random variable, the probability of success (changeover) is: \[P(X_i = 1) = P(\text{changeover between i and i+1}) = p(1-p) + (1-p)p = 2p(1-p)\] The probability of failure (no changeover) is: \[P(X_i = 0) = P(\text{no changeover between i and i+1}) = p^2 + (1-p)^2\]

Step 2: Find the Expectation of Each Random Variable

Now find the expectation of each Bernoulli random variable: \[E(X_i) = 1\cdot P(X_i = 1) + 0\cdot P(X_i = 0) = 2p(1-p)\]

Step 3: Compute the Expected Number of Changeovers

Let Y be the total number of changeovers. Then, Y can be defined as the sum of Bernoulli random variables: \[Y = X_1 + X_2 + \cdots + X_{n-1}\] Then, the expected total number of changeovers is: \[E(Y) = E(X_1 + X_2 + \cdots + X_{n-1})\] Using the linearity of expectation, we have: \[E(Y) = E(X_1) + E(X_2) + \cdots + E(X_{n-1})\] Since all \(X_i\) have the same distribution, the expected value for all is the same: \[E(Y) = (n-1)E(X_1) = (n-1)2p(1-p)\] Hence, the expected number of changeovers in n independent flips of a coin with probability p of landing head is \((n-1)2p(1-p)\).

Key Concepts

Bernoulli Random Variables

When we talk about probabilities, one of the fundamental concepts is the Bernoulli random variable. It is the simplest kind of random variable and is used to model situations with exactly two outcomes: success (often marked as 1) and failure (marked as 0). Flipping a coin, for example, is a classic Bernoulli experiment where 'heads' might be considered a success and 'tails' a failure, or vice-versa.

Each Bernoulli random variable has a single parameter, denoted as \( p \), representing the probability of success. Consequently, \( 1-p \) represents the probability of failure. The outcome of a Bernoulli trial is completely dependent on this probability \( p \). In mathematical terms, for a Bernoulli random variable \( X \), the expectation, or mean, is given by \( E(X) = 1\cdot p + 0\cdot(1-p) = p \).

The exercise mentioned uses Bernoulli random variables to represent the occurrence of changeovers in a sequence of coin flips, which is a creative application of this concept.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random events and determining the likelihood of various outcomes. The foundational building block of probability theory is an event's probability, which ranges from 0 to 1, with 0 indicating an impossibility and 1 signifying certainty.

Probability theory uses a set of axioms and laws, such as the addition and multiplication rules, to derive probabilities of complex events from simpler ones. One crucial aspect of probability is the notion of independence. Events are independent if the outcome of one does not influence the outcome of another, which is a critical concept used in the solution to calculate the expected number of changeovers.

Linearity of Expectation

The linearity of expectation is a powerful property in probability that allows us to say the expected value of the sum of random variables is equal to the sum of their expected values, regardless of whether the variables are independent or not.

In simpler words, if you have random variables \( X_1, X_2, ..., X_n \), then \( E(X_1 + X_2 + ... + X_n) = E(X_1) + E(X_2) + ... + E(X_n) \). This property is especially useful because it allows us to break down complex problems into simpler parts that are easier to solve individually. In the exercise, we apply the linearity of expectation to find the expected number of changeovers by summing the expected values of the Bernoulli random variables that each represent a potential changeover between flips.

Independent Events

Independent events are foundational to probability theory. Two events are independent if the occurrence of one does not affect the probability of the other occurring. Imagine flipping a coin; no matter how many times you flip it, the chance of getting heads remains the same each time - each flip is independent of the previous ones.

Understanding independent events is critical when dealing with multiple random processes, such as the coin flips in our exercise. Since each coin flip does not influence the others, we can calculate probabilities for each flip (or changeover) without worrying about what happened on the previous flips. This independence is crucial for applying the linearity of expectation to find the expected number of changeovers in a sequence of flips.