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Chapter 6: Problem 76

Twenty-five percent of the customers of a grocery store use an express checkout. Consider five randomly selected customers, and let \(x\) denote the number among the five who use the express checkout. a. Calculate \(p(2),\) that is, \(P(x=2)\). b. Calculate \(P(x \leq 1)\). c. Calculate \(P(x \geq 2)\). (Hint: Make use of your answer to Part (b).) d. Calculate \(P(x \neq 2)\).

Short Answer

Expert verified
Based on the given question and solution, the short answer is: a. \(P(x=2) = 0.2637\) b. \(P(x \leq 1) = 0.6328\) c. \(P(x \geq 2) = 0.3672\) d. \(P(x \neq 2) = 0.7363\)

Step by step solution

01

Identify variables of the binomial distribution

: We first need to identify the variables for the binomial distribution formula, which are: - n: Number of trials (in this case, n=5 randomly selected customers). - x: Number of successes we are interested in (number of customers using the express checkout). - p: Probability of a successful trial (25% or 0.25 customers using the express checkout). - q: Probability of an unsuccessful trial (75% or 0.75 customers not using the express checkout).
02

Apply the binomial distribution formula

: To calculate the probability, we apply the binomial distribution formula, which is: \[P(x) = \binom{n}{x}p^xq^{(n-x)}\] a. Calculate the probability P(x=2): For this part, we are looking for the probability that exactly 2 out of 5 customers use the express checkout. Thus, x = 2. \(P(x=2) = \binom{5}{2}(0.25)^2(0.75)^3\)
03

Calculate the binomial coefficient

: The binomial coefficient is calculated using: \[\binom{n}{x} = \frac{n!}{x!(n-x)!}\] In our case \(n=5\) and \(x=2\): \[\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{120}{2*6} = 10\]
04

Calculate the probabilities

: Now, we can plug in the values and calculate the required probabilities: a. Calculate P(x=2): \[P(x=2) = 10*(0.25)^2(0.75)^3 = 0.2637\] Thus, the probability that exactly 2 out of 5 customers use the express checkout is 0.2637. b. Calculate P(x鈮�1): To do this, we have to calculate the probability of P(x=0) and P(x=1) and sum them: \[P(x \leq 1) = P(x = 0) + P(x = 1)\] \[P(x=0) = \binom{5}{0}(0.25)^0(0.75)^5 = 0.2373\] \[P(x=1) = \binom{5}{1}(0.25)^1(0.75)^4 = 0.3955\] \[P(x \leq 1) = 0.2373 + 0.3955 = 0.6328\] Thus, the probability that at most 1 out of 5 customers uses the express checkout is 0.6328. c. Calculate P(x鈮�2): Since P(x鈮�2) is the complement of P(x鈮�1), we can find it by subtracting P(x鈮�1) from 1: \[P(x \geq 2) = 1 - P(x \leq 1) = 1 - 0.6328 = 0.3672\] Thus, the probability that at least 2 out of 5 customers use the express checkout is 0.3672. d. Calculate P(x鈮�2): Since P(x鈮�2) is the complement of P(x=2), we can find it by subtracting P(x=2) from 1: \[P(x \neq 2) = 1 - P(x = 2) = 1 - 0.2637 = 0.7363\] Thus, the probability that a different number than 2 out of 5 customers use the express checkout is 0.7363.

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Key Concepts

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

Binomial Distribution Formula
Understanding the binomial distribution is integral to grasping various probability scenarios, especially when evaluating discrete outcomes. The formula to calculate probabilities in a binomial distribution is denoted as:
\[P(x) = \binom{n}{x}p^xq^{(n-x)}\]
Here's how each component contributes to the formula:
  • \(n\) represents the number of trials or events.
  • \(x\) represents the number of successful outcomes you're seeking.
  • \(p\) is the probability of a single success.
  • \(q\) is the probability of a single failure, which equals \(1-p\).
By applying this formula, you can evaluate situations where there are two outcomes such as pass/fail, yes/no, or success/failure, which makes it a powerful tool in probability theory. In our exercise, we used this formula to model the scenario where customers use or do not use the express checkout at a grocery store.
Probability Calculations
To execute probability calculations using the binomial distribution, follow these logical steps:
  • Identify and substitute the values of \(n\), \(x\), \(p\), and \(q\) into the binomial formula.
  • Calculate the binomial coefficient, which might require factorial computations.
  • Compute each term separately before multiplying them together for precision and ease of understanding.
For example, when we looked for the probability that exactly 2 out of 5 customers use an express checkout, we computed the binomial coefficient for \(n=5\) and \(x=2\), raised the probability of success and failure to their respective powers, and then multiplied them together. This systematic approach reduces the chance of making a mistake and helps in validating the steps when dealing with more complex probability questions.
Binomial Coefficient
The binomial coefficient is a fundamental component of calculating probabilities in binomial distributions. It dictates the number of ways that \(x\) successes can occur in \(n\) trials, regardless of order, and is defined by the formula:
\[\binom{n}{x} = \frac{n!}{x!(n-x)!}\]
Where \(!\) signifies the factorial operation, i.e., the product of an integer and all the positive integers below it. In our exercise, to find the probability of seeing 2 customers using the express checkout out of 5, we computed the binomial coefficient for \(x=2\) and \(n=5\), resulting in:\[\binom{5}{2} = 10\].
This calculation is essential because it provides the number of distinct ways the successes can be arranged, which directly impacts the overall probability result. Overall, mastery of finding the binomial coefficient is vital for accurate probability calculations in binomial scenarios.

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