A contractor estimates the probabilities for the number of...

A contractor estimates the probabilities for the number of days required to complete a certain type of construction project as follows: $$ \begin{array}{lccccc} \hline \text { Time (days) } & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Probability } & 0.05 & 0.20 & 0.35 & 0.30 & 0.10 \\ \hline \end{array} $$ a. What is the probability that a randomly chosen project will take less than 3 days to complete? b. Find the expected time to complete a project. c. Find the standard deviation of time required to complete a project. d. The contractor's project cost is made up of two parts-a fixed cost of $$\$ 20,000$$, plus $$\$ 2,000$$ for each day taken to complete the project. Find the mean and standard deviation of total project cost. e. If three projects are undertaken, what is the probability that at least two of them will take at least 4 days to complete, assuming independence of individual project completion times?

A contractor estimates the probabilities for the number of days required to complete a certain type of construction project as follows:
$$
\begin{array}{lccccc}
\hline \text { Time (days) } & 1 & 2 & 3 & 4 & 5 \\
\hline \text { Probability } & 0.05 & 0.20 & 0.35 & 0.30 & 0.10 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen project will take less than 3 days to complete?
b. Find the expected time to complete a project.
c. Find the standard deviation of time required to complete a project.
d. The contractor's project cost is made up of two parts-a fixed cost of $$\$ 20,000$$, plus $$\$ 2,000$$ for each day taken to complete the project. Find the mean and standard deviation of total project cost.
e. If three projects are undertaken, what is the probability that at least two of them will take at least 4 days to complete, assuming independence of individual project completion times?

Key Concepts

Binomial Distribution and Probabilities

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. This concept is used to compute probabilities of a specified number of events occurring, such as the likelihood of a number of projects meeting a particular time threshold based on independent probabilities.

Linear Transformation of Random Variables

A linear transformation involves altering a random variable by scaling it (multiplying by a constant) and/or shifting it (adding a constant). It adjusts the mean and variance accordingly, which is crucial when modeling costs, earnings, or any dependent measurements derived from the original variable.

Variance and Standard Deviation

Variance measures the spread of a random variable's possible outcomes around its expected value, and the standard deviation is the square root of the variance. These metrics quantify the amount of variability or dispersion in the data, indicating how much the outcomes deviate from the mean.

Cumulative Probability

Cumulative probability refers to the probability that a random variable will take a value less than or equal to a given level. It involves summing the probabilities of outcomes up to a specific point and is often used to answer questions about likelihoods in ranges.

Discrete Probability Distribution

This is a function that provides the probabilities of occurrence of different possible outcomes for a discrete random variable. It allows us to understand how probabilities are assigned to each outcome and forms the basis for calculating other properties like the expected value and variance.

Expected Value

The expected value, or mean, of a random variable is a measure of the central tendency and represents the weighted average of all possible values. It is computed by multiplying each outcome by its probability and summing the results, providing an estimate of the long-run average outcome.

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