Question

A coin that comes up heads with probability \(p\) is continually flipped until the pattern \(\mathrm{T}, \mathrm{T}, \mathrm{H}\) appears. (That is, you stop flipping when the most recent flip lands heads, and the two immediately preceding it lands tails.) Let \(X\) denote the number of flips made, and find \(E[X]\).

Short answer

The expected value of the number of coin flips made until the TTH pattern appears is \(E[X] = \frac{3(1-p)^2 p}{1-(1-p)^3}\).

Step-by-step solution

Note the probabilities

First, let's clearly define the probabilities for the event. \(P(H) = p\) \(P(T) = 1 - p\) Now, let's define the random variable \(X\) as the number of flips made until the pattern TTH appears. #Step 2: Calculate the probability for 3 flips#

Calculate the probability for the first 3 flips

After one flip, the outcome doesn't matter, so we will focus on calculating the probability for the first 3 flips. The probability of getting the TTH pattern in the first 3 flips is: \(P(X=3) = (1-p)^2 p\) #Step 3: Calculate the probability for more than 3 flips#

Calculate the probability for more than 3 flips

Now, let's consider the case where the pattern TTH occurs after more than 3 flips. We can divide this case into 2 sub-cases: Sub-case 1: The last flip is tail (TTT) Probability: \((1-p)^3\) Sub-case 2: The last flip is head (HTH or TTH) Probability: \((1-p)^2 p\) #Step 4: Write down the total probability in terms of conditional probabilities#

Express the conditional probabilities

We can write the total probability for the pattern TTH happening after \(n+1\) flips conditionally based on when TTH happened on the \(n^{th}\) flip: \(P(X = n+1 | X = n) =\) \(P(\text{Sub-case 1}) \times P(\text{Sub-case 2})=\) \((1-p)^3 \times (1-p)^2 p = (1-p)^5 p\) #Step 5: Calculate the expected value of X#

Apply the total expectation theorem

Now we can apply the total expectation theorem to find the expected value of the number of flips made: \(E[X] = \sum_{n=3}^{\infty} n \times P(X = n)\) Using the conditional probabilities derived in step 4, the expected value becomes: \(E[X] = 3(1-p)^2 p + \sum_{n=3}^{\infty} (n+1)(1-p)^5 p\) By evaluating the infinite sum, we get: \(E[X] = \frac{3(1-p)^2 p}{1-(1-p)^3}\) This is the expected value of the number of coin flips made until TTH appears.

Key concepts

Conditional Probability

Understanding conditional probability is essential when analyzing situations where one event's occurrence depends on another. In the context of our exercise, conditional probability allows us to calculate the likelihood of the pattern 'TTH' occurring, given that a certain number of flips have already been made.

To put it simply, if we flip a coin several times, getting a tails (T) followed by another tails and then a heads (H) on the third flip doesn't just happen out of the blue—it depends on what came before. So, if we want to know the probability of TTH happening on the 4th, 5th, 6th flip, and so on, we analyze it knowing what has already occurred; this is where conditional probability steps in. It's like asking 'What are the odds now, given what we've seen so far?'

The formula used in probability theory for conditional probability is typically written as
\( P(A|B) = \frac{P(A \cap B)}{P(B)} \)
where \( P(A|B) \) is the probability of event A given that B has occurred, and \( P(A \cap B) \) is the probability of both A and B happening.

Random Variable

A random variable is a backbone concept in probability theory. It's a rule that assigns a numerical value to each potential outcome in a space of possible outcomes. Think of it as a way to keep score in the game of chance—how many flips till we see 'TTH'? With each toss of the coin, the random variable, denoted by \( X \) in our exercise, keeps track of the flip count until we hit our pattern.

There are two types of random variables: discrete and continuous. Our exercise deals with a discrete random variable because the number of coin flips counts in whole numbers—you can't flip a coin halfway. To calculate the expected value (or mean) of such a variable, we multiply each possible value by its probability, then sum all these products. It's like figuring out the average number of times you'll need to flip that coin to see 'TTH' pop up.

Probability Theory

Probability theory gives us a systematic way to study uncertainty and randomness. It's the mathematical framework that underpins games of chance, statistics, and even machine learning. In this exercise, we use probability theory to tackle questions like: 'What's the chance of flipping a tail?', 'What's the likelihood of flipping a head after two tails?', and 'How many flips on average to get the TTH sequence?'.

Fundamental to this theory are the concepts of events, outcomes, and probabilities. An 'event' is just a set of outcomes—for example, getting 'TTH' is an event that can occur in various ways over multiple coin flips. Each possible sequence of flips leading to 'TTH' is an outcome, and probability theory allows us to assign each outcome a numerical probability, so we can make predictions about how likely different sequences are.

Total Expectation Theorem

The Total Expectation Theorem, sometimes called the Law of Total Expectation, is a powerful tool that simplifies the calculation of expected values by breaking down complex random processes into simpler ones. It's particularly useful when a problem involves several stages or layers of random events.

In practical terms, you first identify all the possible scenarios or conditions your random variable can experience. For each, you calculate the expected value given that scenario. Then, you weigh each expected value by the probability of its corresponding scenario and add them up to get the overall expected value.

For example, in the coin flip exercise, we don't just consider the chance of getting 'TTH' in a vacuum; we look at the different ways in which flips can progress and conditionally calculate the expected number of flips until 'TTH' happens. This theorem incredibly simplifies the process by allowing us to treat each potential path to 'TTH' separately before bringing them together for the big picture.