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Chapter 6: Problem 29

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. \(n=6, p=0.3\)

Short Answer

Expert verified
Mean: 1.8, SD: 1.1225. Histogram: Right-skewed.

Step by step solution

01

- Understand the Parameters

Identify the given parameters for the binomial distribution: the number of trials, which is \( n = 6 \), and the probability of success in each trial, which is \( p = 0.3 \).
02

- Construct the Binomial Probability Distribution

Use the binomial probability formula: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \) for \( k = 0, 1, 2, 3, 4, 5, 6 \). Calculate each probability and construct the distribution table.
03

- Calculate Each Probability

For each value of \( k \): \( \binom{6}{k} (0.3)^k (0.7)^{6-k} \): \( P(X = 0) = \binom{6}{0} (0.3)^0 (0.7)^6 = 0.117649 \) \( P(X = 1) = \binom{6}{1} (0.3)^1 (0.7)^5 = 0.302526 \) \( P(X = 2) = \binom{6}{2} (0.3)^2 (0.7)^4 = 0.324135 \) \( P(X = 3) = \binom{6}{3} (0.3)^3 (0.7)^3 = 0.18522 \) \( P(X = 4) = \binom{6}{4} (0.3)^4 (0.7)^2 = 0.059535 \) \( P(X = 5) = \binom{6}{5} (0.3)^5 (0.7)^1 = 0.010206 \) \( P(X = 6) = \binom{6}{6} (0.3)^6 (0.7)^0 = 0.000729 \). Table these values.
04

- Compute the Mean

The mean (expected value) of a binomial distribution is given by \( \text{Mean} = \text{E}(X) = np \). Substitute the parameters: \( \text{E}(X) = 6 \times 0.3 = 1.8 \).
05

- Compute the Standard Deviation

The standard deviation of a binomial distribution is given by \( \text{SD}(X) = \text{sqrt}(np(1-p)) \). Substitute the parameters: \( \text{SD}(X) = \text{sqrt}(6 \times 0.3 \times 0.7) = \text{sqrt}(1.26) \rightarrow 1.1225 \).
06

- Draw the Probability Histogram

Using the probabilities calculated for each \( k \) value, plot a histogram. The horizontal axis represents the number of successes (\( k \)), and the vertical axis represents the probabilities. Mark the mean (1.8) on the histogram.
07

- Comment on the Histogram Shape

The histogram likely has a right-skewed shape due to the higher probability of fewer successes. This is common when \( p < 0.5 \).

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Key Concepts

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

Probability Distribution
In a binomial distribution, we're dealing with scenarios where there are a fixed number of independent trials, each with two possible outcomes: success or failure. The probability of success remains the same for each trial. The binomial probability distribution shows the probability of having exactly k successes in n trials. For example, with parameters n = 6 (trials) and p = 0.3 (probability of success), we calculate the probability for each possible number of successes (0 to 6) using the binomial formula.
Mean of Binomial Distribution
The mean of a binomial distribution is the expected number of successes over the given trials and is calculated using the formula: \[ \text{Mean} = np \]. Here, 'n' is the number of trials, and 'p' is the probability of success in each trial. For our example, with n = 6 and p = 0.3, the mean is: \[ \text{Mean} = 6 \times 0.3 = 1.8 \]. This tells us that on average, we can expect 1.8 successes out of 6 trials.
Standard Deviation of Binomial Distribution
The standard deviation gives us a measure of how spread out the values of a binomial distribution are around the mean. It can be calculated using the formula: \[ \text{SD}(X) = \text{sqrt}(np(1-p)) \]. For our parameters n = 6 and p = 0.3, the standard deviation is calculated as: \[ \text{SD}(X) = \text{sqrt}(6 \times 0.3 \times 0.7) = \text{sqrt}(1.26) \rightarrow 1.1225 \]. This value tells us the extent to which the number of successes deviates from the mean on average.
Probability Histogram
A probability histogram visually represents the binomial distribution. On the horizontal axis, we plot the number of successes (k), and on the vertical axis, we plot the corresponding probabilities derived from the binomial formula. Using the calculated probabilities: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] for each k (0 through 6), we construct bars for each probability value. The height of each bar correlates with the probability for each number of successes. When p < 0.5, like in our example, the histogram tends to be right-skewed, showing higher probabilities for fewer successes.
Binomial Formula
The binomial formula is crucial for calculating the exact probabilities of having a specific number of successes in a given number of trials. It is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]. \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose k successes from n trials. For example, to find the probability of exactly 2 successes out of 6 trials with p = 0.3, we compute: \[ P(X = 2) = \binom{6}{2} (0.3)^2 (0.7)^4 \]. This formula helps build the complete probability distribution, as seen in our example's detailed steps.

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Most popular questions from this chapter

In the Cash Five Lottery in Connecticut, a player pays \(\$ 1\) for a single ticket with five numbers. Five Ping-Pong balls numbered 1 through 35 are randomly chosen from a bin without replacement. If all five numbers on a player's ticket match the five chosen, the player wins \(\$ 100,000 .\) The probability of this occurring is \(\frac{1}{324,632} .\) If four numbers match, the player wins \(\$ 300 .\) This occurs with probability \(\frac{1}{2164^{\circ}}\). If three numbers match, the player wins \(\$ 10 .\) This occurs with probability \(\frac{1}{75} .\) Compute and interpret the expected value of the game from the player's point of view.

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