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Chapter 5: Problem 55

If \(x\) is a binomial random variable, calculate the probability of \(x\) for each case. a. \(\quad n=4, x=1, p=0.3\) b. \(\quad n=3, x=2, p=0.8\) c. \(\quad n=2, x=0, p=\frac{1}{4}\) d. \(\quad n=5, x=2, p=\frac{1}{3}\) e. \(\quad n=4, x=2, p=0.5\) f. \(\quad n=3, x=3, p=\frac{1}{6}\)

Short Answer

Expert verified
The attempted probabilities would be as followed depending on the case a-f for which the calculations were shown in step 1-6. Remember that all probabilities obtained should fall between 0 (least possible case) and 1 (most possible case).

Step by step solution

01

- Case a: Calculate \(P(X=1)\) where \(n=4, p=0.3\)

First, plug in the given values into the Binomial Probability formula :\(P(X=1) = \binom{4}{1} \cdot (0.3)^{1} \cdot (1 - 0.3)^{4-1}\)Calculate the above expression to get the probability.
02

- Case b: Calculate \(P(X=2)\) where \(n=3, p=0.8\)

Plug in the given values into the Binomial Probability formula :\(P(X=2) = \binom{3}{2} \cdot (0.8)^{2} \cdot (1 - 0.8)^{3-2}\)Calculate the above expression to get the probability.
03

- Case c: Calculate \(P(X=0)\) where \(n=2, p=\frac{1}{4}\)

Input the values into the formula :\(P(X=0)= \binom{2}{0} \cdot (\frac{1}{4})^{0} \cdot (1 - \frac{1}{4})^{2-0}\)Compute the above expression to obtain the probability.
04

- Case d: Calculate \(P(X=2)\) where \(n=5, p=\frac{1}{3}\)

Plug in the given values into the Binomial Probability formula :\(P(X=2) = \binom{5}{2} \cdot (\frac{1}{3})^{2} \cdot (1 - \frac{1}{3})^{5-2}\)Calculate the above expression to get the probability.
05

- Case e: Calculate \(P(X=2)\) where \(n=4, p=0.5\)

Plug in the values into the formula :\(P(X=2) = \binom{4}{2} \cdot (0.5)^{2} \cdot (1 - 0.5)^{4-2}\)Compute the above expression to obtain the probability.
06

- Case f: Calculate \(P(X=3)\) where \(n=3, p=\frac{1}{6}\)

Plug in the given values into the Binomial Probability formula :\(P(X=3) = \binom{3}{3} \cdot (\frac{1}{6})^{3} \cdot (1 - \frac{1}{6})^{3-3}\)Calculate the above expression to get the probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Distribution
The binomial distribution is a fundamental concept in statistics that models the number of successes in a fixed number of independent Bernoulli trials. Each trial has only two possible outcomes: success or failure. For instance, flipping a coin results in heads (success) or tails (failure).

The probability of exactly 'x' successes in 'n' trials, where the probability of success on a single trial is 'p', is given by the binomial probability formula:
\[ P(X=x) = \binom{n}{x} p^x (1 - p)^{n-x} \]
where \( \binom{n}{x} \) represents the number of ways to choose 'x' successes out of 'n' trials, known as 'combinatorial' number or 'combination'. This formula was applied to the textbook exercise for various values of 'n', 'x', and 'p'.
Probability Theory
Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides the mathematical foundation to assess risk, make predictions, and inform decisions in every day scenarios as well as in complex scientific studies.

In the context of the binomial distribution, probability theory helps us to understand the behavior of a process where results are binary, such as the flipping of a coin or answering a true/false question. By using probability theory, students are equipped with the tools to calculate the likelihood of various outcomes and to make sense of random events in a structured and quantifiable way.
Combinatorics

Understanding Combinations

Combinatorics involves the study of counting, arranging, and combination of elements according to specified rules. The textbook exercise uses a fundamental combinatorial concept called 'combinations', denoted by \( \binom{n}{x} \), where 'n' is the total number of items to choose from, and 'x' is the number of items being chosen, without regard to order.

These combinations are critical in calculating binomial probabilities. For example, when finding the number of ways to get 2 heads in 4 coin flips, combinations tell us how many ways we can pick 2 flips out of 4 to be heads - which is \( \binom{4}{2} \). Understanding the role of combinations allows students to compute probabilities correctly in binomial distributions.
Random Variables
In probability and statistics, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. In the given exercise, 'x' represents a discrete random variable that can take on a specific set of values - the number of successful outcomes in binomial trials.

Random variables are categorized as either 'discrete' (taking on a countable number of distinct values) or 'continuous' (taking on any value within an interval or collection of intervals). The values of the random variable in the binomial distribution are discrete since we count the number of successes which can only be integers from 0 to 'n'. Understanding random variables is essential to analyzing different types of data and their distributions.

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Most popular questions from this chapter

Extended to overtime in game 7 on the road in the 2002 NBA play-offs, the two- time defending champion Los Angeles Lakers did what they do best-thrived when the pressure was at its highest. Both of the Lakers' star players had their chance at the foul line late in overtime. a. With 1: 27 minutes left in overtime and the game tied at \(106-106,\) Shaquille (Shaq) O'Neal was at the line for two free-throw attempts. He has a history of making 0.555 of his free-throw attempts, and during this game, prior to these two shots, he had made 9 of his 13 attempts. Justify the statement "The law of averages was working against him." Both players made both shots, and the series with the Sacramento Kings was over. b. With 0: 06 seconds left in overtime and the game score standing at \(110-106,\) Kobe Bryant was at the line for two free-throw shots. He has a history of making 0.829 of his free throws, and during this game, prior to these two shots, he had made 6 of his 8 attempts. Justify the statement "The law of averages was working for him."

a. Form the probability distribution table for \(P(x)=\frac{x}{6},\) for \(x=1,2,3.\) b. Find the extensions \(x P(x)\) and \(x^{2} P(x)\) for each \(x.\) c. \(\quad\) Find \(\Sigma[x P(x)]\) and \(\Sigma\left[x^{2} P(x)\right].\) d. Find the mean for \(P(x)=\frac{x}{6},\) for \(x=1,2,3.\) e. Find the variance for \(P(x)=\frac{x}{6},\) for \(x=1,2,3.\) f. Find the standard deviation for \(P(x)=\frac{x}{6},\) for \(x=1,2,3.\)

A carton containing 100 T-shirts is inspected. Each T-shirt is rated "first quality" or "irregular." After all 100 T-shirts have been inspected, the number of irregulars is reported as a random variable. Explain why \(x\) is a binomial random variable.

a. Explain the difference and the relationship between a probability distribution and a probability function. b. Explain the difference and the relationship between a probability distribution and a frequency distribution, and explain how they relate to a population and a sample.

A binomial random variable has a mean equal to 200 and a standard deviation of \(10 .\) Find the values of \(n\) and \(p.\)
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