Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.4.75

Chapter 11, Problem 11.4.75

Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)

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Recognize that the expected number of tosses to get the first head in a fair coin toss is given by the infinite series \(\sum_{k=1}^{\infty} k \left(\frac{1}{2}\right)^k\). This is a weighted sum where \(k\) represents the toss number and \(\left(\frac{1}{2}\right)^k\) is the probability of getting the first head on the \(k\)-th toss.

Recall the geometric series formula: for \(|x| < 1\), \(\sum_{k=0}^{\infty} x^k = \frac{1}{1-x}\). To involve the factor \(k\) in the sum, differentiate both sides with respect to \(x\) to bring down the \(k\) term.

Differentiate the geometric series term-by-term: \(\frac{d}{dx} \sum_{k=0}^{\infty} x^k = \sum_{k=1}^{\infty} k x^{k-1} = \frac{d}{dx} \left( \frac{1}{1-x} \right) = \frac{1}{(1-x)^2}\). Multiply both sides by \(x\) to get \(\sum_{k=1}^{\infty} k x^k = \frac{x}{(1-x)^2}\).

Substitute \(x = \frac{1}{2}\) into the differentiated series formula to evaluate the sum: \(\sum_{k=1}^{\infty} k \left(\frac{1}{2}\right)^k = \frac{\frac{1}{2}}{\left(1 - \frac{1}{2}\right)^2}\).

Simplify the expression to find the expected number of tosses required to get the first head. This value represents the average number of tosses needed in the long run.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Expected Value in Probability

The expected value is the average outcome of a random experiment over many trials. For discrete random variables, it is calculated as the sum of each possible value multiplied by its probability. In this problem, it represents the average number of coin tosses needed to get the first head.

Geometric Series and Its Sum

A geometric series is a sum of terms where each term is a constant ratio times the previous term. The sum of an infinite geometric series with ratio |r| < 1 is S = a / (1 - r), where a is the first term. Recognizing the series in the problem as geometric helps in evaluating it.

Differentiation of a Geometric Series

Differentiating the sum formula of a geometric series with respect to its ratio allows us to find sums involving terms multiplied by their indices, like ∑ k r^k. This technique is key to evaluating the expected value series, which includes the term k(1/2)^k.

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