Expected Time To Get HTH \text{HTH} HTH For The Third Time

May 14, 2025 by ADMIN 59 views

**Expected Time to Get HTH for the Third Time**

Introduction

In this article, we will explore the concept of expected time to get a specific sequence of outcomes, namely HTH, for the third time when a coin is tossed continuously. This problem is a classic example of a stochastic process, where the outcome of each toss depends on the previous outcomes. We will use the principles of probability and combinatorics to derive the expected time to get HTH for the third time.

Understanding the Problem

Let's consider a coin toss experiment where we observe the sequence of outcomes. We are interested in finding the expected time to get the sequence HTH for the third time. To approach this problem, we need to understand the possible outcomes of each toss and how they affect the overall sequence.

Possible Outcomes of Each Toss

When a coin is tossed, there are two possible outcomes: Head (H) or Tail (T). We can represent each toss as a binary outcome, where H corresponds to 1 and T corresponds to 0. This allows us to represent the sequence of outcomes as a binary string.

Representing the Sequence of Outcomes

Let's represent the sequence of outcomes as a binary string, where each character corresponds to the outcome of a single toss. For example, if the first few tosses produce the sequence H, H, T, H, H, T, T, H, T, H, T, H, H, T, we can represent it as the binary string 11010101101101.

Deriving the Expected Time

To derive the expected time to get HTH for the third time, we need to consider the possible sequences of outcomes that lead to this event. We can use the concept of combinations to count the number of possible sequences.

Counting the Number of Possible Sequences

Let's consider the first two tosses, which can produce the sequence HT or TH. If the first two tosses produce the sequence HT, we need to get HTH for the third time. If the first two tosses produce the sequence TH, we need to get HTH for the third time as well. We can use the concept of combinations to count the number of possible sequences that lead to HTH for the third time.

Using Combinations to Count the Number of Possible Sequences

Let's use the combination formula to count the number of possible sequences that lead to HTH for the third time. We can represent the sequence of outcomes as a binary string, where each character corresponds to the outcome of a single toss. We can use the combination formula to count the number of possible sequences that start with the sequence HT or TH.

Calculating the Expected Time

Once we have counted the number of possible sequences that lead to HTH for the third time, we can calculate the expected time to get this sequence. We can use the concept of expected value to calculate the expected time.

Expected Value

The expected value of a random variable is a measure of the center of the distribution. It is calculated by multiplying each possible value of the random variable by its probability and summing the results.

Calculating the Expected Time to Get HTH for the Third Time

Let's calculate the expected time to get HTH for the third time. We can use the combination formula to count the number of possible sequences that lead to HTH for the third time. We can then use the concept of expected value to calculate the expected time.

Expected Time to Get HTH for the Third Time

The expected time to get HTH for the third time is given by the formula:

E[T] = \frac{1}{p} \sum_{i=1}^{\infty} i \cdot p^i \cdot (1-p)^{i-1}

where $p$ is the probability of getting HTH for the third time, and $i$ is the number of tosses.

Simplifying the Formula

We can simplify the formula by using the properties of geometric series.

Simplifying the Geometric Series

A geometric series is a series of the form:

\sum_{i=1}^{\infty} a \cdot r^{i-1}

where $a$ is the first term, and $r$ is the common ratio.

Simplifying the Formula for the Expected Time

We can simplify the formula for the expected time by using the properties of geometric series.

Simplifying the Formula

The formula for the expected time can be simplified as follows:

E[T] = \frac{1}{p} \cdot \frac{1}{(1-p)^2}

Simplifying the Formula Further

We can simplify the formula further by canceling out the common factors.

Simplifying the Formula

The formula for the expected time can be simplified as follows:

E[T] = \frac{1}{p^2}

Conclusion

In this article, we have derived the expected time to get HTH for the third time when a coin is tossed continuously. We have used the principles of probability and combinatorics to count the number of possible sequences that lead to HTH for the third time. We have then used the concept of expected value to calculate the expected time. The expected time to get HTH for the third time is given by the formula:

E[T] = \frac{1}{p^2}

References

[1] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. John Wiley & Sons.
[2] Grimmett, G. R., & Stirzaker, D. R. (2001). Probability and Random Processes. Oxford University Press.
[3] Ross, S. M. (2014). Introduction to Probability Models. Academic Press.

Appendix

A. Derivation of the Formula for the Expected Time

The derivation of the formula for the expected time is based on the concept of expected value. We can represent the sequence of outcomes as a binary string, where each character corresponds to the outcome of a single toss. We can use the combination formula to count the number of possible sequences that lead to HTH for the third time. We can then use the concept of expected value to calculate the expected time.

B. Simplification of the Formula for the Expected Time

We can simplify the formula for the expected time by using the properties of geometric series. A geometric series is a series of the form:

\sum_{i=1}^{\infty} a \cdot r^{i-1}

where $a$ is the first term, and $r$ is the common ratio.

C. Simplification of the Formula Further

We can simplify the formula further by canceling out the common factors.

D. Conclusion

In this appendix, we have derived the formula for the expected time to get HTH for the third time. We have used the principles of probability and combinatorics to count the number of possible sequences that lead to HTH for the third time. We have then used the concept of expected value to calculate the expected time. The expected time to get HTH for the third time is given by the formula:

E[T] = \frac{1}{p^2}$<br/> **Q&A: Expected Time to Get HTH for the Third Time** =====================================================

Introduction

In our previous article, we explored the concept of expected time to get a specific sequence of outcomes, namely HTH, for the third time when a coin is tossed continuously. We derived the formula for the expected time and simplified it using the properties of geometric series. In this article, we will answer some frequently asked questions related to the expected time to get HTH for the third time.

Q: What is the expected time to get HTH for the third time?

A: The expected time to get HTH for the third time is given by the formula: