Conditions For Business Decisions Using Subjective Probabili

Conditions for Business Decisions Using Subjective Probability

Subjective probability is used in business when decisions are made based on personal judgment, experience, or intuition rather than purely mathematical analysis or historical data. This approach is often employed in situations where data is scarce, unreliable, or unavailable, and decision-makers must rely on their own assessment of the likelihood of various outcomes.

For example, a company considering entering a new market might estimate the probability of success based on industry knowledge and past experiences, even without concrete data. Similarly, a manager might gauge the likelihood of a competitor launching a new product based on rumors or market signals, rather than hard statistics. These subjective probabilities are inherently personal and can vary between individuals but are essential when objective data cannot be sourced or is insufficient.

Explaining Bayes’ Theorem in Plain Language

Bayes’ theorem provides a way to update our beliefs about how likely something is, given new evidence. In the equation P(H/D) = [P(H) × P(D/H)] / [P(H) × P(D/H) + P(~H) × P(D/~H)], the goal is to find the probability that a hypothesis H is true, given the observed data D.

P(H) is called the prior probability. It reflects our initial belief about the likelihood of hypothesis H before seeing the new data. P(D/H) is the probability of observing the data D if H is true—this is called the likelihood. P(~H) represents the probability that H is not true (the complement of H), and P(D/~H) is the probability of observing the data D if H is false.

The entire equation essentially combines these components to give an updated probability of H after considering the new evidence D. It allows us to refine our beliefs based on observed data, which is crucial in decision-making, diagnostics, or predictive modeling.

Understanding Probability Trees Through an Example

Probability trees serve as visual tools to map out all possible outcomes of a sequence of probabilistic events. When flipping a coin, since each flip has a 50% chance of landing heads or tails, the probability tree helps compute the probabilities of different combinations of outcomes.

In the student's example, the probability of flipping heads twice is calculated as 0.5 × 0.5 = 0.25, representing the combined likelihood of getting heads on both flips. Extending this to all outcomes, such as heads-tails, tails-heads, and tails-tails, the total probability sums up to 1, covering all possibilities. Probability trees simplify the process of calculating complex joint probabilities by breaking them down into sequential steps.

Modeling Business Outcomes and Calculating Expected Profit and Variability

For the construction firm bidding on a contract, the random variable X can represent the profit outcome, with different values corresponding to different scenarios. Specifically:

• If the firm wins the full contract, profit = +50,000
• If it wins a shared contract, profit = +20,000
• If it wins nothing, profit = 0

The probabilities associated with these outcomes are: 20% (0.20) for full contract, 75% (0.75) for shared contract, and the remaining 5% (0.05) for no contract.

The expected profit is calculated by multiplying each outcome by its probability and summing these values, which offers an estimate of the average profit over many similar bidding situations. The standard deviation reflects the variability or risk associated with the profit outcomes, indicating how much the actual results might deviate from the average.

Characteristics of Standard Normal Distribution and Data in Organizations

The standard normal distribution is characterized by its symmetric bell shape centered around the mean, with most data clustered around the average and fewer extremes at the tails. It has a mean of zero and a standard deviation of one, making it a benchmark for many statistical analyses.

In an organization, data such as employee age or salary might approximate a normal distribution, especially if the sample size is large and the data is naturally spread around an average. For example, employee ages often follow a normal distribution because most employees cluster around a typical working age, with fewer very young or very old employees. Conversely, categorical data like gender or ethnicity typically do not follow normal distribution, as they are qualitative variables with discrete categories.

Calculating Threshold Weight for M&M's Packages

Given that M&M's weigh 0.86 grams on average with a standard deviation of 0.04 grams, the coefficient of variation is approximately 4.7%, indicating low relative variability. To ensure that a package contains at least 60 pieces with 99.5% certainty, we need to determine the weight threshold corresponding to the 99.5th percentile of the distribution.

Since the weight per package is normally distributed, we can calculate the cutoff weight by adding the mean to the z-score associated with the 99.5% percentile (which is approximately 2.81 for the standard normal distribution) multiplied by the standard deviation. This ensures that 99.5% of packages will weigh more than this threshold, thus containing at least 60 pieces with high confidence.

References

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