Question

1. imagine youre managing a software development project. you want to estimate the probability of successfully completing the project within a given time frame, based on past experiences. you have observed the following:

• in similar projects, 70% of the time, they were completed successfully within the deadline.
• youve also tracked the outcomes and found that in 8 out of 10 similar projects, success was achieved.

suppose want to know the probability that the project will succeed with a probability greater than 0.7.

a) find \\(\alpha\\) and \\(\beta\\).

b) set up the pdf needed to find this probability and then set up the integral.

c) use r or some other technology to determine the probability.

Explanation:

Identify prior parameters

Using the Beta Distribution knowledge point

$$
LATEXBLOCK0
$$

Update parameters with observed data

Using the Beta-Binomial Conjugacy and Bayesian Updating knowledge points

$$
LATEXBLOCK1
$$

Formulate probability density function and integral

Using the Beta Distribution knowledge point

$$
LATEXBLOCK2
$$

Calculate final probability

Using the Beta Distribution knowledge point

$$
LATEXBLOCK3
$$

Answer:

a)

\(\alpha = 15\) and \(\beta = 5\)

b)

The probability density function is:

$$f(p) = \frac{\Gamma(20)}{\Gamma(15)\Gamma(5)} p^{14} (1-p)^4 = 38760 p^{14} (1-p)^4, \quad 0 \le p \le 1$$

The integral to find the probability that \(p > 0.7\) is:

$$P(p > 0.7) = \int_{0.7}^{1} 38760 p^{14} (1-p)^4 dp$$

c)

Using R code 1 - pbeta(0.7, 15, 5), the probability is approximately \(0.6730\) (or \(67.30\%\)).