Chapter 3: Problem 24
. Let
Short Answer
Expert verified
The pmf of X, which represents the number of tosses required to get two consecutive heads in a series of independent toss-fair-coins, can be viewed as a Fibonacci series where each Fibonacci number is divided by . Furthermore, the sum of the probabilities for all possible outcomes determined according to this pmf equals to one, thus confirming its validity as a probability mass function.
Step by step solution
01
Illustrate Connection to Fibonacci Series
Firstly, observe the similarity between the pmf given and a Fibonacci series. The number of ways two consecutive heads can occur in a sequence can indeed be represented by a Fibonacci series, where gives the number of sequences of length ending in tails and the number of sequences of length ending in heads. When is replaced with , we get the Fibonacci series
02
Calculate Probabilities Using the pmf
In this step, calculate the probability as described using the relationship with the Fibonacci series. As the outcomes are independent, the probability of each outcome can be multiplied directly: , where and denote the probabilities of obtaining heads and tails, respectively.
03
Represent Fibonacci Series as a Mathematical Function
Recall the formula given in the second part of the question, which represents the nth Fibonacci number, , as a mathematical function. This formula implies , and this will be used in the next step to normalize the pmf.
04
Verify Normality of the pmf
To confirm the pmf, we need to demonstrate that it adds up to 1 over all possible outcomes, which is a property of any valid probability distribution. Evaluate using the Fibonacci series formula, which is equivalent to substituting with the expression from Step 3. After mathematical simplification, this arithmetic series sums to 1, showing the normality of the pmf under consideration.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Fibonacci Series
The Fibonacci series is a captivating sequence where each number is the sum of the two preceding ones, typically starting with 0 and 1. It is defined by the recurrence relation:
with the initial conditions . This series appears in different areas of mathematics and science, often surprising beginners with its frequent and sometimes unexpected appearances. For instance, in our context, the Fibonacci series comes into play when determining the count of scenarios leading to consecutive heads in coin tosses.
with the initial conditions
Link to Probability
When looking at probability through the lens of the Fibonacci series, it becomes evident that the series can represent possible outcomes. In the problem presented, the series corresponds to the number of ways to get two consecutive heads, taking into account the length of tosses.Practical Appearance
Beyond abstract applications, the Fibonacci series also appears in nature. Many plants display patterns that conform to the Fibonacci sequence, such as the arrangement of leaves on a stem or the pattern of scales on a pinecone. It's a fascinating bridge between mathematics and the natural world.It's important for students to connect the properties of the Fibonacci series to real-world contexts to improve retention and deepen understanding. Recognizing such patterns can make the abstract more tangible and, consequently, easier to grasp.Exploring Consecutive Heads Probability
The concept of calculating the probability of getting consecutive heads in a series of coin tosses is a classic problem in probability theory. Underlying the rules of independence and combination, the probability mass function (pmf) associates a specific probability with the number of coin tosses required to observe the first occurrence of consecutive heads.
In the simplest terms, the probability of getting heads on a single toss of a fair coin is 0.5 or . To expand on this, the probability of getting two heads in a row would be since the tosses are independent and the occurrence of heads in each toss is multiplied. ) is tied to the Fibonacci series through a formula. By understanding this connection, one can reason out the chances of each successive toss leading to the desired outcome. As students internalize this relationship, calculating such probabilities becomes significantly more intuitive.
In the simplest terms, the probability of getting heads on a single toss of a fair coin is 0.5 or
Understanding Through Recursion
Here's where it gets more interesting. When considering longer sequences, we must account for every possible way that we could arrive at two consecutive heads, whether it ended in tails or heads in the earlier toss. This is inherently recursive, much like the Fibonacci sequence itself.Application to the Exercise
In the exercise, the probability for each number of tosses (Delving into Mathematical Statistics
Mathematical statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. It provides us with tools to make sense of data, decide on the level of uncertainty, and make predictions.
In terms of our exercise, mathematical statistics involves the probability theory to express and analyze the process of observing consecutive heads. The pmf is an essential concept within statistics that assigns probabilities to discrete random variables—values that result from a random phenomenon, which in our case is the coin toss.
In terms of our exercise, mathematical statistics involves the probability theory to express and analyze the process of observing consecutive heads. The pmf is an essential concept within statistics that assigns probabilities to discrete random variables—values that result from a random phenomenon, which in our case is the coin toss.