Probability Calculation Example

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Questions and Answers

What does the equation change to after extracting values unaffected by the limit?

  • A quadratic equation
  • An inverse of the original equation
  • An expanded version of the RHS of the equation (correct)
  • A completely different equation

What happens to each value in equation (e) if N becomes infinitely large?

  • They become negative
  • They remain unchanged
  • They double in value
  • They tend towards 1 (correct)

How are the numerator and denominator values in equation (d) described?

  • Simply powers of R (correct)
  • Exponential functions
  • Polynomials in terms of N
  • Square roots of R

What operation is performed on equation (f) after the 'limit'?

<p>Subtraction (B)</p> Signup and view all the answers

In the context of the text, what do we need to do to work towards the next probability distribution?

<p>Finish the expansion of (b) (B)</p> Signup and view all the answers

What is the formula for calculating the total number of possible combinations in the scenario described?

<p>$NCR = 5!/(3!2!)$ (A)</p> Signup and view all the answers

What is the probability calculation formula for finding the overall probability of R successes in N trials?

<p>$P(R successes in N trials) = NCR * p^{R} * q^{N-R}$ (C)</p> Signup and view all the answers

What is the name of the probability distribution described in the text?

<p>Binomial Probability Distribution (D)</p> Signup and view all the answers

In the context provided, what does 'Binomial' refer to?

<p>Two possible state of nature outcomes (A)</p> Signup and view all the answers

What does p represent in the probability formula mentioned?

<p>Probability of achieving R successes (C)</p> Signup and view all the answers

Considering the scenario, what is the probability of making fewer than three sales?

<p>$P(&lt;3 sales) = P(0 sales) + P(1 sale) + P(2 sales)$ (A)</p> Signup and view all the answers

What aspect is important to consider when using the Poisson distribution function according to Black (2004)?

<p>Having a constant rate of success (A)</p> Signup and view all the answers

According to Black (2004), what does the variation in the rate of success λ imply for the Pizza Manager's staffing rota?

<p>It necessitates variable staffing levels (A)</p> Signup and view all the answers

How is the use of Poisson distribution function in Business often categorized according to Black (2004)?

<p>As Queuing Theory (A)</p> Signup and view all the answers

Why is managing arrival queues or departing goods important in a business context?

<p>To minimize bottlenecks (C)</p> Signup and view all the answers

What does the Poisson distribution function allow managers to visualize according to Black (2004)?

<p>'Successes' against a continuous background (D)</p> Signup and view all the answers

In what context has the Poisson distribution function found particular value according to Black (2004)?

<p>Minimizing bottlenecks in organizational functions (D)</p> Signup and view all the answers

What is the probability of no sales success in the given scenario?

<p>0.015625 (C)</p> Signup and view all the answers

What is the probability of exactly 3 sales successes?

<p>0.3125 (C)</p> Signup and view all the answers

What is the probability of no sales, 1 sale, or 2 sales?

<p>0.67 (C)</p> Signup and view all the answers

If 34 out of 54 students in the 2011 class are sampled, and the probability of a student feeling they received 'good' support is 0.75, what is the expected number of students who will express that they received 'good' support?

<p>25.5 (B)</p> Signup and view all the answers

What is the probability that a student felt they received 'good' support during their studies?

<p>0.75 (D)</p> Signup and view all the answers

If the probability of a student feeling they received 'good' support is 0.75, what is the probability of a student feeling they did not receive 'good' support?

<p>0.25 (C)</p> Signup and view all the answers

What is the predicted probability of getting 2 questions incorrect according to the table?

<p>0.39 (A)</p> Signup and view all the answers

What does the student's predicted probability distribution sum up to?

<p>1.0 (A)</p> Signup and view all the answers

What does the phrase 'long run average' refer to in the context of the given information?

<p>The average number of incorrect answers across all students in the class (B)</p> Signup and view all the answers

Which formula is given for calculating the mean (μ) of a discrete probability distribution?

<p>Both a and c (C)</p> Signup and view all the answers

Based on the information given, what can be inferred about the student's confidence level in scoring less than 5 questions incorrectly?

<p>The student is less confident about scoring less than 5 questions incorrectly (B)</p> Signup and view all the answers

What is the meaning of the symbol Σ used in the formulas for calculating the mean?

<p>It represents summation or the sum of a series (A)</p> Signup and view all the answers

What is the formula for calculating the probability of R successes in N trials?

<p>(pR * qN-R) (D)</p> Signup and view all the answers

In the context of the text, what does P((N-R) trials being failures) = qN-R represent?

<p>Probability of failures in N trials (B)</p> Signup and view all the answers

What does P(3 in ALL spins (or R=5)) = p5 represent?

<p>Probability of 3 in all spins (A)</p> Signup and view all the answers

How is the relationship between subsequent independent events calculated?

<p>Using the AND operator (A)</p> Signup and view all the answers

What is the probability of seeing our chosen number 3 times in 5 spins?

<p>(p3 * q3) (C)</p> Signup and view all the answers

What does the formula pR represent in the given text?

<p>Probability of R successes (C)</p> Signup and view all the answers

What is the key challenge in explaining the Poisson sequence to managers and students of business?

<p>Lack of understanding its linkage to the Binomial distribution (C)</p> Signup and view all the answers

What is the primary focus when discussing the Poisson sequence in practical organizational contexts according to the text?

<p>Highlighting its role in managing and allocating resources (C)</p> Signup and view all the answers

How do most students transition between the Binomial and Poisson distribution functions according to the text?

<p>By relying on faith and context matching (D)</p> Signup and view all the answers

Why is a detailed explanation of the origin of the Poisson distribution important according to the text?

<p>To clarify its value to managers (B)</p> Signup and view all the answers

What is a common challenge faced by students when dealing with the Poisson sequence?

<p>Inability to match contexts to interpretation (A)</p> Signup and view all the answers

In what year did Simeon-Denis Poisson publish the essence of the Poisson sequence formulation?

<p>1837 (D)</p> Signup and view all the answers

According to the passage, which of the following situations would NOT be well-suited for using the Poisson distribution function?

<p>Predicting the number of defective products in a manufacturing process with a high defect rate (B)</p> Signup and view all the answers

According to Black (2004), what is a crucial assumption when using the Poisson distribution function?

<p>The rate of success (λ) remains constant over the time period of interest (B)</p> Signup and view all the answers

In the context of the Poisson distribution, what does the parameter λ represent?

<p>The expected number of successes in a given time period or area (D)</p> Signup and view all the answers

Which of the following statements best describes the Poisson distribution's applicability in a business context, according to the passage?

<p>It is useful for analyzing customer behavior and operational processes with low success rates (D)</p> Signup and view all the answers

If the rate of success λ varies significantly over the time period of interest, what would be the appropriate action according to Black (2004)?

<p>Use a different probability distribution model, as the Poisson distribution is not suitable (A)</p> Signup and view all the answers

Based on the information provided, which of the following statements is NOT true about the Poisson distribution?

<p>It is applicable when the probability of success is high and the probability of failure is low (D)</p> Signup and view all the answers

What is the fundamental assumption made about the probability of a number appearing or not appearing in a single spin of the roulette wheel?

<p>The probability of a number appearing or not appearing remains constant, regardless of the outcomes of previous spins. (B)</p> Signup and view all the answers

If P represents the probability of a desired number appearing, and Q represents the probability of it not appearing, what is the relationship between P and Q for a single spin of the roulette wheel?

<p>P + Q = 1 (A)</p> Signup and view all the answers

If N represents the number of spins of the roulette wheel, and R represents the number of times the desired number appears, what is the probability of exactly R successes (the desired number appearing) in N spins?

<p>P(exactly R successes) = P(R successes) × P(N-R failures) (B)</p> Signup and view all the answers

If we let P represent the probability of success (the desired number appearing) and Q represent the probability of failure (the desired number not appearing), what is the relationship between P and Q?

<p>Q = 1 - P (C)</p> Signup and view all the answers

If the desired number is 3, and we want to find the probability of it appearing exactly 2 times in 5 spins of the roulette wheel, which of the following expressions would we use?

<p>$P(2 \text{ successes}) = \binom{5}{2} p^2 q^3$ (C)</p> Signup and view all the answers

Suppose we want to find the probability of the desired number appearing at most twice in 10 spins of the roulette wheel. Which of the following expressions would we use?

<p>$P(\text{at most 2 successes}) = \sum_{r=0}^{2} \binom{10}{r} p^r q^{10-r}$ (D)</p> Signup and view all the answers

What is the probability of getting exactly 3 sales successes out of 10 trials, if the probability of success on each trial is 0.3?

<p>$\binom{10}{3} \times 0.3^3 \times 0.7^7$ (C)</p> Signup and view all the answers

If the probability of making a sale on each trial is p, what is the expression for the probability of making exactly 2 sales out of 5 trials?

<p>$\binom{5}{2} \times p^2 \times (1-p)^3$ (D)</p> Signup and view all the answers

If the probability of making a sale on each trial is 0.4, what is the probability of making at least 2 sales out of 6 trials?

<p>$\binom{6}{2} \times 0.4^2 \times 0.6^4 + \binom{6}{3} \times 0.4^3 \times 0.6^3 + \binom{6}{4} \times 0.4^4 \times 0.6^2 + \binom{6}{5} \times 0.4^5 \times 0.6^1 + \binom{6}{6} \times 0.4^6 \times 0.6^0$ (C)</p> Signup and view all the answers

If the probability of making a sale on each trial is p, what is the expression for the probability of making no sales out of n trials?

<p>$\binom{n}{0} \times p^0 \times (1-p)^n$ (A)</p> Signup and view all the answers

What does the parameter lambda represent in the context of the Poisson distribution?

<p>The rate of success or the average number of successes in a fixed interval</p> Signup and view all the answers

How is the relationship between P (probability of success) and Q (probability of failure) defined in a single spin of the roulette wheel?

<p>P + Q = 1</p> Signup and view all the answers

What is the formula for calculating the probability of R successes in N trials?

<p>$P(R) = \binom{n}{r} p^{r} q^{n-r}$</p> Signup and view all the answers

What is the primary focus when discussing the Poisson sequence in practical organizational contexts according to the text?

<p>Modeling rare events or occurrences</p> Signup and view all the answers

How would you describe the distribution function and queuing theory in the context of business management?

<p>Tools used for rational decision-making and optimizing operational processes</p> Signup and view all the answers

Define discrete random variables and provide an example of such a variable.

<p>Discrete random variables are data values that come from decision contexts and are presented as positive integer data. For example, the number of students with black hair in a sample of 6 students selected randomly from a class of 30.</p> Signup and view all the answers

Explain the significance of the Poisson distribution in practical organizational contexts.

<p>The Poisson distribution allows managers to visualize the occurrence of rare events or arrivals over a period of time, aiding in decision-making and resource allocation.</p> Signup and view all the answers

Differentiate between discrete and continuous variables in the context of mathematical functions.

<p>Discrete variables are fully describable outcomes that are usually positive integer data, while continuous variables have an infinite number of possible outcomes within a given range.</p> Signup and view all the answers

How does the Poisson distribution function assist in managing arrival queues or departing goods in a business context?

<p>The Poisson distribution helps in modeling and predicting the arrival rates of customers or goods, allowing businesses to efficiently manage resources and optimize operations.</p> Signup and view all the answers

Explain the primary focus when discussing the Binomial distribution in practical organizational contexts.

<p>The primary focus when discussing the Binomial distribution is on scenarios where there are a fixed number of trials, each with a binary outcome (success or failure), with a constant probability of success.</p> Signup and view all the answers

What is the name of the French statistician who the Poisson distribution is named after, and when did he publish the essence of the formulation?

<p>Simeon-Denis Poisson in 1837</p> Signup and view all the answers

What is the fundamental assumption about the probability of a number appearing or not appearing in a single spin of the roulette wheel?

<p>Independent events</p> Signup and view all the answers

What does the symbol λ represent in the context of the Poisson distribution?

<p>Rate of success</p> Signup and view all the answers

What is a crucial assumption when using the Poisson distribution function?

<p>Constant rate of success over time</p> Signup and view all the answers

What is the meaning of the symbol Σ used in the formulas for calculating the mean?

<p>Summation</p> Signup and view all the answers

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