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Chapter 4: Problem 29

Let \(y_{1}

Short Answer

Expert verified
The algorithm has a probability of 50% of selecting the largest value.

Step by step solution

01

Understand the setup

First release is the task at hand is based on three ordered numbers \(y_{1}<y_{2}<y_{3}\) which are given out of order. The aim is to pick the largest number on a single attempt using a specific algorithm.
02

Examine the algorithm

The algorithm adopted involves initially ignoring the first number, then choosing the second if it's greater than the first, or else choosing the third. Identifying the strategy helps in understanding why the success ratio might be 50%.
03

Determine permutations

Consider all six possible orderings of the observed values \( y_{1}< y_{2}< y_{3} \), which are presented in an arbitrary order: \( y_{1} y_{2} y_{3}, y_{1} y_{3} y_{2}, y_{2} y_{1} y_{3}, y_{2} y_{3} y_{1}, y_{3} y_{1} y_{2}, y_{3} y_{2} y_{1}\). The algorithm executes differently based on each scenario.
04

Calculate probability

In three of these situations, the largest number, \( y_{3}\), would be selected. They include: \( y_{2} y_{3} y_{1}, y_{1} y_{3} y_{2}, y_{3} y_{1} y_{2}\). Thus, the likelihood of making the right choice is \( \frac{3}{6}= \frac{1}{2}\), or 50%.
05

Conclusion

This demonstrates that the supposed algorithm has a probability of 50% of selecting the largest value.

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Most popular questions from this chapter

Let \(X\) and \(Y\) denote independent random variables with respective probability density functions \(f(x)=2 x, 0

Let \(\bar{X}\) denote the mean of a random sample of size 25 from a gamma-type distribution with \(\alpha=4\) and \(\beta>0 .\) Use the Central Limit Theorem to find an approximate \(0.954\) confidence interval for \(\mu\), the mean of the gamma distribution. Hint: \(\quad\) Use the random variable \((\bar{X}-4 \beta) /\left(4 \beta^{2} / 25\right)^{1 / 2}=5 \bar{X} / 2 \beta-10\).

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Let \(Y_{1}

Let \(Y_{1}
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