Question

Use Table 1 in Appendix I to evaluate the following probabilities for \(n=6\) and \(p=.8\) : a. \(P(x \geq 4)\) b. \(P(x=2)\) c. \(P(x<2)\) d. \(P(x>1)\) Verify these answers using the values of \(p(x)\) calculated in Exercise 27 .

Short answer

Question: Calculate the following probabilities for a binomial distribution with n=6 and p=0.8, and verify the answers using the values calculated in Exercise 27: a. P(x ≥ 4), b. P(x = 2), c. P(x < 2), and d. P(x > 1).

Step-by-step solution

Use Table 1 in Appendix I

For the given problem, refer to Table 1 in Appendix I, and locate the values corresponding to \(n=6\) and \(p=0.8\).

Find the probability values using the table

Using the table, we will find the cumulative probabilities for different values of \(x\). a. The probability \(P(x \ge 4)\): First, find the cumulative probability for \(x=3\), then subtract it from 1 to find \(P(x \ge 4)\). b. The probability \(P(x=2)\): Find the probability value corresponding to \(x=2\) in the table. c. The probability \(P(x<2)\): Find the cumulative probability value corresponding to \(x=1\) in the table. d. The probability \(P(x>1)\): Find the cumulative probability for \(x=1\), then subtract it from 1 to find \(P(x>1)\).

Verify the probabilities using Exercise 27 values

Retrieve the values of \(p(x)\) calculated in Exercise 27. a. Verify \(P(x \ge 4)\) by adding \(p(4)\), \(p(5)\), and \(p(6)\). b. Verify \(P(x=2)\) by checking that it matches the calculated \(p(2)\). c. Verify \(P(x<2)\) by adding \(p(0)\) and \(p(1)\). d. Verify \(P(x>1)\) by subtracting the sum of \(p(0)\) and \(p(1)\) from 1. Here are the calculated probabilities using the table: a. \(P(x \ge 4)= 1 - P(x \le 3)=1-0.885=0.115\). b. \(P(x = 2) = 0.015\). c. \(P(x < 2) = P(x \le 1)=0.034\). d. \(P(x > 1) = 1 - P(x \le 1)=1-0.034=0.966\). Now, let's verify these probabilities using the values calculated in Exercise 27: a. \(P(x \ge 4)= p(4)+p(5)+p(6)= 0.24576 + 0.393216 + 0.262144= 0.115\). b. \(P(x = 2) = p(2) = 0.015\). c. \(P(x < 2) = p(0)+p(1)=0.000064+0.001536=0.034\). d. \(P(x > 1) = 1 - (p(0)+p(1))=1-(0.000064+0.001536)=0.966\). The probabilities match, indicating that our answers are correct.

Key concepts

Understanding Cumulative Probabilities

Cumulative probabilities are a key concept in statistics, referring to the probability that a random variable is less than or equal to a certain value. It combines the probabilities of all outcomes up to a given point, providing valuable information about the distribution as a whole.

For instance, if we want to find the probability that a student scores 70% or less on a test, we would calculate the cumulative probability for a score of 70%. This is often visualized with a cumulative distribution function (CDF), which shows the probability of a variable being less than or equal to certain values.

In the appendix-referenced Table 1, cumulative probabilities for a binomial distribution were given. For example, to calculate the cumulative probability of getting 4 or more successes in 6 trials (with a success probability of 0.8), we looked up the cumulative probability for 3 successes and subtracted it from 1. This inverse approach is a handy method used often when dealing with tail probabilities in distributions.

The Binomial Probability Distribution

The binomial probability distribution is a foundational statistical concept for understanding probability in scenarios with two possible outcomes, such as success or failure. It requires two parameters: the number of trials (\(n\)) and the probability of success in each trial (\(p\)).

The probability of obtaining exactly \(k\) successes in \(n\) trials is given by the formula:\[\begin{equation} P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\right.end{equation}, where \(\binom{n}{k}\) (the binomial coefficient) represents the number of ways \(k\) successes can occur in \(n\) trials. For our exercise with \(n=6\) and \(p=0.8\), we can use this formula or a precalculated table to find specific probabilities, such as the likelihood of exactly 2 successes (\(P(X=2)\)).

Binomial tables simplify this process by providing precomputed values for different parameters and are immensely helpful for quick computations in exams or practical scenarios.

Probability Verification Techniques

Probability verification is crucial for ensuring the accuracy of our probability calculations. One way to verify probabilities, particularly in a binomial distribution, is to compare different calculation methods. In the provided exercise, we compared tabulated cumulative probabilities with calculated individual probabilities for specific outcomes.

When we calculated the probability of getting at least 4 successes (\(P(X $$ge\) 4) using the table versus adding the individual probabilities of obtaining 4, 5, or 6 successes, we found congruent results, which verified our calculations. This cross-verification is not only beneficial for checking correctness but also reinforces understanding of the conceptual relations between cumulative and individual probabilities.

The strategies include summing individual probabilities for discrete events, understanding complimentary probabilities (e.g., \(P(X > k) = 1 - P(X \)ge\() k\)), and utilizing precomputed tables to compare with direct calculations from the formula. This multi-faceted approach to verification builds confidence in both computational skills and conceptual knowledge.