Problem 99
Question
A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n},\) where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).
Step-by-Step Solution
Verified Answer
The sum of the series in part (a) is 1. The series in part (b) is not geometric. The sum of the series in part (b) can be found using a computer algebra system.
1Step 1: Finding whether series in (a) is geometric
Observe that the series \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}\) can be rewritten as \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...\). This suggests the ratio \(r\) between consecutive terms is \(1/2\) as each term is half the previous term.
2Step 2: Computing sum in (a)
Given that series is geometric with first term \(a = 1/2\) and \(r = 1/2\), the sum \(s\) of this infinite series is given by \(s = a / (1 - r)\). Substituting values, \(s = (1/2) / (1 - 1/2) = 1\). Hence, \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n} = 1\)
3Step 3: Finding nature of series in (b)
Look at the series: \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\), Each term could be formed by multiplying \(n\) with \(\left(\frac{1}{2}\right)^n.\) Although each term is halved before proceeding to the next, the \(n\) outside the parentheses breaks the ratio characteristic of a geometric series. So, this is not a geometric series.
4Step 4: Calculating sum in (c)
Using a computer algebra system, input the series \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) to compute its sum. Depending upon the system, the method may vary, but typically involves inputting the series using the system’s syntax for infinity and sigma notation.
Key Concepts
Geometric SeriesInfinite SeriesExpected ValueComputational Tools
Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In our problem concerning a fair coin toss, we saw the series \[\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\]This series is geometric because the ratio of each term to the previous term is \(1/2\).
The formula for the sum \(s\) of an infinite geometric series with first term \(a\) and common ratio \(|r| < 1\) is \(s = \frac{a}{1 - r}\).
In this case, \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\), leading to the sum being \[s = \frac{1/2}{1 - 1/2} = 1\]This confirms that the infinite series sums to 1.
The formula for the sum \(s\) of an infinite geometric series with first term \(a\) and common ratio \(|r| < 1\) is \(s = \frac{a}{1 - r}\).
In this case, \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\), leading to the sum being \[s = \frac{1/2}{1 - 1/2} = 1\]This confirms that the infinite series sums to 1.
Infinite Series
An infinite series is a sum of an infinite sequence of terms. In mathematics, we often encounter infinite series when dealing with continuous processes that repeat indefinitely.
Let's consider our example: the sum of all probabilities of getting the first head on the \(n\)th coin toss. It's expressed as \[\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}\]This is an infinite series because it includes an endless number of terms.
To understand why the sum of this series converges, or has a finite value, it's crucial to recognize the behavior of geometric series. When the common ratio \(r\) is a fraction between -1 and 1, the terms get smaller, approaching zero as \(n\) increases, allowing the sum to settle at a finite number.
Let's consider our example: the sum of all probabilities of getting the first head on the \(n\)th coin toss. It's expressed as \[\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}\]This is an infinite series because it includes an endless number of terms.
To understand why the sum of this series converges, or has a finite value, it's crucial to recognize the behavior of geometric series. When the common ratio \(r\) is a fraction between -1 and 1, the terms get smaller, approaching zero as \(n\) increases, allowing the sum to settle at a finite number.
Expected Value
The expected value in probability theory represents the average outcome if an experiment is repeated many times. It provides a measure of the center of the probability distribution.
For our coin toss problem, the expected number of tosses needed to get the first head is given by a series:\[\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\]Here, each term in the series is formed by multiplying the number of tosses \(n\) by the probability of it being the first head.
However, this series is not geometric because the terms involve an additional multiplication by \(n\), altering the constant ratio pattern characteristic of geometric series.
Calculating the expected value involves finding the sum of this series, which can yield insights into the average number of attempts required.
For our coin toss problem, the expected number of tosses needed to get the first head is given by a series:\[\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\]Here, each term in the series is formed by multiplying the number of tosses \(n\) by the probability of it being the first head.
However, this series is not geometric because the terms involve an additional multiplication by \(n\), altering the constant ratio pattern characteristic of geometric series.
Calculating the expected value involves finding the sum of this series, which can yield insights into the average number of attempts required.
Computational Tools
With the complexity of some mathematical series like the expected value in our problem, manual computation can be cumbersome. Modern computational tools such as computer algebra systems (CAS) can facilitate this process.
These tools allow input of mathematical definitions using predefined syntax, handling operations like summation over infinite ranges. To find the sum of the series \[\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\]a CAS can quickly compute the result.
These tools allow input of mathematical definitions using predefined syntax, handling operations like summation over infinite ranges. To find the sum of the series \[\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\]a CAS can quickly compute the result.
- This can include entering commands in software like Mathematica, Maple, or even Python libraries such as SymPy.
- These utilities provide accuracy and efficiency, especially helpful for checking homework or verifying manual calculations.
Other exercises in this chapter
Problem 98
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