Question

A coin having probability \(p\) of coming up heads is successively flipped until two of the most recent three flips are heads. Let \(N\) denote the number of flips. (Note that if the first two flips are heads, then \(N=2 .\) ) Find \(E[N]\).

Short answer

The expected number of flips E[N] needed to have two of the most recent three coin flips be heads for a coin with probability p of landing heads is given by the expression: \(E[N] = 5p^2 - 3p^4\).

Step-by-step solution

Define the probability of each scenario happening

Let's define the probabilities for the scenarios we mentioned earlier: 1. Probability of scenario A (HH): \(P(A) = p^2\) 2. Probability of scenario B (HTH): \(P(B) = p(1-p)p\) 3. Probability of scenario C (THH): \(P(C) = (1-p)pp\)

Calculate the conditional expected values

Now, we need to find the conditional expected values for each scenario: 1. E[N|A]: If the first two flips are heads, the number of flips required is 2. 2. E[N|B]: If we have HTH, the number of flips required is 3. 3. E[N|C]: If we have THH, the number of flips required is 3.

Calculate the expected value E[N] using the law of total expectation

Now we'll use the law of total expectation to find the expected number of flips E[N]. The law of total expectation is defined as: \(E[N] = E[N|A]P(A) + E[N|B]P(B) + E[N|C]P(C)\) Plugging in the values we found in Steps 1 and 2: \(E[N] = 2p^2 + 3p(1-p)p + 3(1-p)pp\) Now, we'll simplify this expression: \(E[N] = 2p^2 + 3p^2(1-p) + 3(1-p)p^2\)

Simplify the expression and find E[N]

Now let's simplify E[N] further by combining terms: \(E[N] = 2p^2 + 3p^3 - 3p^4 + 3p^2 - 3p^3\) Combining terms: \(E[N] = 2p^2 + 3p^3 - 3p^4 + 3p^2 - 3p^3\) \(E[N] = 5p^2 - 3p^4\) This is the expected number of flips E[N] needed to have two of the most recent three coin flips be heads for a coin with probability p of landing heads.

Key concepts

Probability Models

In studying random phenomena, a probability model is a mathematical representation of a random process that outlines all possible outcomes and assigns probabilities to those outcomes. For example, a fair coin has two possible outcomes: heads or tails, each with a probability of 0.5.

In the given exercise, the probability model is slightly more complex. Here, we're dealing with a sequence of coin flips with the likelihood of landing heads being given by the probability p. The range of outcomes is more extensive because we're looking at sequences of flips rather than individual flips. To construct this model, we categorize sequences of flips into scenarios, each with an associated probability. By doing this, we analyze a random process that extends over several trials—an essential concept in probability that gives rise to chain events, leading to a final outcome only after a series of experiments.

An understanding of probability models is crucial because it sets the foundation for computing expected values, analyzing patterns over time, and predict the behavior of more complicated random processes, like the one in the exercise.

Law of Total Expectation

The law of total expectation, also known as the tower rule, is a fundamental principle in probability theory that allows us to break down complex expected values into simpler components. This law states that the expected value of a random variable can be computed by taking the average of its conditional expectations, weighted by the probabilities of the corresponding conditions.

In the context of the given exercise, we are asked to find the expected number of coin flips, E[N]. We approach this by first finding the expected number of flips conditional on each scenario: two consecutive heads (A), a head-tail-head sequence (B), and a tail-head-head sequence (C). Once we have these conditional expected values, we apply the law of total expectation to combine them, taking into account the probability of each scenario. This breakdown not only simplifies the calculation but also provides deep insight into how each possible sequence of events contributes to the overall expected number of flips.

Understanding the law of total expectation is beneficial for students as it enables them to handle complex expectations by dissecting a problem into understandable segments, offering a clearer view on how different probabilities influence the expected outcome of random variables.

Conditional Expected Value

A conditional expected value is the expected value of a random variable given that a certain condition or set of conditions is satisfied. It's like taking an average where we only include the outcomes that meet specific criteria. This concept helps to focus on particular segments of a probability distribution, allowing for a detailed analysis of complex random processes.

In our coin-flipping scenario, we determine the number of flips needed under specific conditions (two heads in the first two flips, or certain head-tail-head or tail-head-head sequences). These are examples of conditional expected values, denoted as E[N|A], E[N|B], and E[N|C]. Each of these calculated values assumes a different sequence of flips has occurred. By focusing on these conditions, we obtain a clearer picture of how the coin flip sequences unfold and how they affect the overall expectation.

Grasping the concept of conditional expected value is essential. It underscores the role of specific outcomes in the context of their probability and how they collectively influence the expected result of a broader random process. This concept is not only critical to solving probability problems but also to real-world situations where decisions are made based on expected results under certain conditions.