Q: A computer consulting firm presently hasbids out on three projects. Let A_i ={awarded project i}, for i = 1, 2, 3, and supposethat P(A₁) = 0.23,P(A₂) = 0.25,P(A₃) = 0.29,P(A₁ ? A₂) = 0.07,P(A₁ ? A₃) = 0.05,P(A₂ ? A₃) = 0.08,P(A₁ ? A₂ ?A₃) = 0.02. Use the probabilities given aboveto compute the following probabilities, and explain in words themeaning of each one. (Round your answers to four decimalplaces.)

(a)    P(A₂ |A₁) =

Explain this probability in words.

If the firm is awarded project 2, this is the chance they willalso be awarded project 1. If the firm is awarded project 1, thisis the chance they will also be awarded project2.     This is the probability that the firm isawarded either project 1 or project 2. This is the probability thatthe firm is awarded both project 1 and project 2.

(b)    P(A₂ ?A₃ | A₁) =

Explain this probability in words.

This is the probability that the firm is awarded projects 1, 2,and 3. If the firm is awarded project 1, this is the chance theywill also be awarded projects 2 and 3.     Ifthe firm is awarded projects 2 and 3, this is the chance they willalso be awarded project 1. This is the probability that the firm isawarded at least one of the projects.

(c)    P(A₂ ?A₃ | A₁) =

Explain this probability in words.

If the firm is awarded project 1, this is the chance they willalso be awarded at least one of the other two projects. This is theprobability that the firm is awarded at least one of theprojects.     If the firm is awarded at leastone of projects 2 and 3, this is the chance they will also beawarded project 1. This is the probability that the firm is awardedprojects 1, 2, and 3.

(d)    P(A₁ ?A₂ ? A₃ |A₁ ? A₂ ?A₃) =

Explain this probability in words.

This is the probability that the firm is awarded at least one ofthe projects. This is the probability that the firm is awardedprojects 1, 2, and 3.     If the firm isawarded at least two of the projects, this is the chance that theywill be awarded all three projects. If the firm is awarded at leastone of the projects, this is the chance that they will be awardedall three projects.

Q: Three couples and two single individualshave been invited to an investment seminar and have agreed toattend. Suppose the probability that any particular couple orindividual arrives late is 0.38 (a couple will travel together inthe same vehicle, so either both people will be on time or elseboth will arrive late). Assume that different couples andindividuals are on time or late independently of one another. LetX = the number of people who arrive late for theseminar.

(a) Determine the probability mass function of X.[Hint: label the three couples #1, #2, and #3 and the twoindividuals #4 and #5.] (Round your answers to four decimalplaces.)

x	P(X =x)
0
1
2
3
4
5
6
7
8

(b) Obtain the cumulative distribution function of X.(Round your answers to four decimal places.)

x	F(x)
0
1
2
3
4
5
6
7
8

Use the cumulative distribution function of X tocalculate

P(2 ? X ? 7).

(Round your answer to four decimal places.)

P(2 ? X ? 7) =