91Ó°ÊÓ

In probability theory, two events are considered independent if the occurrence of one event does not influence the occurrence of the other. To mathematically determine independence, we use the formula: \[ P(A \cap B) = P(A) \cdot P(B) \]Here, \(P(A \cap B)\) is the probability that both events \(A\) and \(B\) occur at the same time. If this equation holds true, then the events are independent.In the context of the exercise, we are evaluating two contract bids. If obtaining contract \#1 affects the probability of obtaining contract \#2, then these events are not independent. According to the problem, the probability of getting contract \#2 changes based on whether contract \#1 is obtained or not, hence indicating dependency between the two events.

The expected value is a fundamental concept in probability, representing the average outcome one could expect from a random process over the long run. For discrete random variables, like the number of contracts obtained in this exercise, the expected value \(E[X]\) can be calculated using:\[ E[X] = \sum (x_i \cdot P(X=x_i)) \]Where \(x_i\) are the possible outcomes, and \(P(X=x_i)\) are their respective probabilities.In this example, the possible outcomes are getting 0, 1, or 2 contracts. The expected value is determined by multiplying each outcome by its probability and summing the results. This gives us insights into the "average" number of contracts we might receive if the bidding process were repeated many times.

Standard deviation, in probability, measures the amount of variation or dispersion of a set of values. It's based on the variance, which is the average of the squared differences from the Mean. For a probability distribution, it provides insights into how much the outcomes vary from the expected value.To find the standard deviation:1. Calculate the expected value \(E[X]\).2. Compute the variance \(\sigma^2\) using:\[ \sigma^2 = \sum (x_i - E[X])^2 \cdot P(X=x_i) \]3. The standard deviation \(\sigma\) is the square root of the variance:\[ \sigma = \sqrt{\sigma^2} \]In the exercise, this measure differs with each possible outcome of contracts, highlighting how spread out these probabilities are around the expected value.

A probability model provides a mathematical description of a random experiment's possible outcomes and their likelihoods. In the context of the contracts exercise, a probability model helps us understand the probabilities of receiving different numbers of contracts.For example, let \(X\) denote the number of contracts acquired. The probability model lists the possible values of \(X\) (0, 1, or 2 contracts) along with the associated probabilities. These probabilities can be calculated using:- \(P(X=0)\): Probability of getting no contracts.- \(P(X=1)\): Probability of getting exactly one contract.- \(P(X=2)\): Probability of getting both contracts.This model provides a clear framework for anticipating the outcomes of the bid processes, helping in making strategic decisions based on the likelihood of each scenario.