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Magazine Chapter 5: Problem 8
Use the formula for the binomial probability distribution to calculate the values of \(p(x)\), and construct the probability histogram for \(x\) when \(n=6\) and \(p=.2\). [HINT: Calculate \(P(x=k)\) for seven different values of \(k\).]
Short Answer
Expert verified
Answer: The probability of having 3 successes in the experiment is 0.081920.
Step by step solution
01
Understanding the Binomial Probability Formula
The binomial probability formula is: $$P(x=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) (\(n!\) means the factorial of \(n\))
Step 2: Calculate the probabilities for \(x=0, 1, 2, ..., 6\)
02
Calculating Probabilities
In this step, we will plug in the given values \(n=6\) and \(p=0.2\) into the binomial probability formula and find the probability for each value of \(k\). In this case, \(k=0,1,2,\dots, 6\).
$$
P(x=0) = \binom{6}{0} (0.2)^0 (1-0.2)^{6-0} \\
P(x=1) = \binom{6}{1} (0.2)^1 (1-0.2)^{6-1} \\
P(x=2) = \binom{6}{2} (0.2)^2 (1-0.2)^{6-2} \\
P(x=3) = \binom{6}{3} (0.2)^3 (1-0.2)^{6-3} \\
P(x=4) = \binom{6}{4} (0.2)^4 (1-0.2)^{6-4} \\
P(x=5) = \binom{6}{5} (0.2)^5 (1-0.2)^{6-5} \\
P(x=6) = \binom{6}{6} (0.2)^6 (1-0.2)^{6-6}
$$
Step 3: Calculate the binomial coefficients and probabilities
03
Calculating Binomial Coefficients and Probabilities
We will now calculate each binomial coefficient and probability by plugging the values for \(k\) into the formula:
$$
P(x=0) = \frac{6!}{0!(6-0)!} (0.2)^0 (1-0.2)^{6-0} = 0.262144 \\
P(x=1) = \frac{6!}{1!(6-1)!} (0.2)^1 (1-0.2)^{6-1} = 0.393216 \\
P(x=2) = \frac{6!}{2!(6-2)!} (0.2)^2 (1-0.2)^{6-2} = 0.245760 \\
P(x=3) = \frac{6!}{3!(6-3)!} (0.2)^3 (1-0.2)^{6-3} = 0.081920 \\
P(x=4) = \frac{6!}{4!(6-4)!} (0.2)^4 (1-0.2)^{6-4} = 0.015360 \\
P(x=5) = \frac{6!}{5!(6-5)!} (0.2)^5 (1-0.2)^{6-5} = 0.001536 \\
P(x=6) = \frac{6!}{6!(6-6)!} (0.2)^6 (1-0.2)^{6-6} = 0.000064
$$
Step 4: Construct the Probability Histogram
04
Constructing the Probability Histogram
To construct the probability histogram, we will create a bar chart with the calculated probabilities as the heights of the bars. The x-axis will represent the number of successes \(x\) (0 to 6), and the y-axis will represent the probability \(P(x=k)\).
Label the x-axis with the values 0 to 6, and label the y-axis with the probabilities calculated in Step 3. Create a bar for each value of \(x\), where the height of the bar corresponds to its respective probability:
- \(P(x=0) = 0.262144\)
- \(P(x=1) = 0.393216\)
- \(P(x=2) = 0.245760\)
- \(P(x=3) = 0.081920\)
- \(P(x=4) = 0.015360\)
- \(P(x=5) = 0.001536\)
- \(P(x=6) = 0.000064\)
This probability histogram illustrates the distribution of probabilities for each number of successes in a binomial experiment with \(n=6\) trials and a success probability \(p=0.2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Binomial Coefficient
The binomial coefficient is a central concept in probability and statistics, especially within the binomial probability distribution. It is represented using the symbol \( \binom{n}{k} \). This notation expresses the number of ways to choose \( k \) successes from \( n \) trials.
This coefficient is crucial when calculating probabilities for binomial distributions, as it accounts for all possible combinations of \( k \) successes. It is computed using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]- \( n! \) is the factorial of \( n \)- \( k! \) represents the factorial of \( k \)- \( (n-k)! \) is the factorial of the difference between \( n \) and \( k \)
In practice, calculating the binomial coefficient involves understanding the underlying concept of factorial, which we will discuss shortly. Remember that this formula helps determine the number of possible outcomes in which exactly \( k \) events occur out of \( n \) trials.
This coefficient is crucial when calculating probabilities for binomial distributions, as it accounts for all possible combinations of \( k \) successes. It is computed using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]- \( n! \) is the factorial of \( n \)- \( k! \) represents the factorial of \( k \)- \( (n-k)! \) is the factorial of the difference between \( n \) and \( k \)
In practice, calculating the binomial coefficient involves understanding the underlying concept of factorial, which we will discuss shortly. Remember that this formula helps determine the number of possible outcomes in which exactly \( k \) events occur out of \( n \) trials.
Visualizing with a Probability Histogram
A probability histogram is an excellent tool to visualize the distribution of probabilities in a binomial experiment. It provides a graphical representation that helps to interpret and understand the probability of different outcomes. Each bar in the histogram represents one possible number of successes (from 0 to \( n \)), with the height of the bar corresponding to the probability of that outcome.
To construct a probability histogram for a binomial distribution:
Visualizing data in this way makes it easier to see patterns and understand how probable each outcome is. It allows for an intuitive grasp of the distribution, differentiating between more and less likely results.
To construct a probability histogram for a binomial distribution:
- Label the x-axis with the possible outcomes (e.g., 0, 1, 2,... up to the total number of trials)
- Mark the y-axis with the range of probabilities, ensuring it covers from 0 to the highest probability value found
- Draw a bar for each outcome, where the height of the bar is proportional to its probability
Visualizing data in this way makes it easier to see patterns and understand how probable each outcome is. It allows for an intuitive grasp of the distribution, differentiating between more and less likely results.
The Role of Factorial in Calculations
Factorial is a mathematical operation denoted by an exclamation mark \( ! \). It is essential in various areas of mathematics, including the computation of binomial coefficients. The factorial of a number \( n \), represented as \( n! \), is the product of all positive integers less than or equal to \( n \).
For example:
Understanding factorials is crucial when using the formula for the binomial coefficient, \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) because it tells us how many combinations are possible. Being familiar with factorials can simplify your calculations, as it is frequently used to calculate permutations and combinations in probability and statistics. Whether you are computing binomial probabilities manually or interpreting formulas, knowing how to handle factorials is indispensable.
For example:
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 0! = 1 \) by definition
Understanding factorials is crucial when using the formula for the binomial coefficient, \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) because it tells us how many combinations are possible. Being familiar with factorials can simplify your calculations, as it is frequently used to calculate permutations and combinations in probability and statistics. Whether you are computing binomial probabilities manually or interpreting formulas, knowing how to handle factorials is indispensable.
