SOLUTION: A symmetrical coin is tossed n times, with the outcomes marked by the letters H and S. A sequence of n letters is then shortened by crossing out all the H's before the first S and

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Question 1200600: A symmetrical coin is tossed n times, with the outcomes marked by the letters H and S. A sequence of n letters is then shortened by crossing out all the H's before the first S and all the H's after the last S (if any). What is the probability that the shortened sequence of outcomes will be no longer than m letters? ([n, m] = [14, 4])
Answer by CPhill(1959) About Me  (Show Source):
You can put this solution on YOUR website!
Certainly, let's find the probability.
**Understanding the Problem**
* **Coin Tosses:** A fair coin is flipped 14 times (n = 14).
* **Outcome Representation:** Each flip is represented by 'H' (heads) or 'S' (tails).
* **Shortening the Sequence:**
* All 'H's before the first 'S' are removed.
* All 'H's after the last 'S' are removed.
* **Goal:** Determine the probability that the shortened sequence has a length of 4 or less (m = 4).
**Approach**
1. **Consider Possible Scenarios:**
* **All Tails (S...S):** Shortened sequence is 14 'S's.
* **One Tail:**
* 'S' followed by any combination of 'H's and 'S's.
* Shortened sequence is the entire original sequence.
* **Two Tails:**
* 'H...' (any number of 'H's) followed by 'S', followed by any combination of 'H's and 'S's.
* Shortened sequence is limited by the position of the first and last 'S'.
* **Three or Four Tails:** Similar to the two-tails case, the shortened sequence is limited by the positions of the tails.
2. **Calculate Probabilities:**
* We need to calculate the probabilities of each scenario (all tails, one tail, two tails, etc.) and sum them up.
3. **Determine the Probability of a Shortened Sequence of Length 4 or Less**
* Sum the probabilities of all scenarios where the shortened sequence has a length of 4 or less.
**Note:**
* This problem involves combinations and can be solved using probability theory and combinatorics.
* The exact calculation can be quite complex.
**To get the precise probability, you can:**
* **Use a computer program or statistical software:** Implement the logic described above to calculate the probabilities.
* **Utilize a probabilistic programming language:** Languages like Python (with libraries like NumPy) or R can be used to efficiently calculate the probabilities.
I hope this explanation helps! Let me know if you have any further questions.