Probability Mathematics

1. Roll 2 dice. Let A denote the event that the sum of the dices is 7.  Let B denote the event that the first die is 4. Are A and B independent? Justify your answer using probabilities.

 

 

 

 

 

 

 

 

 

 

 

 

2 A balanced die is thrown once.  If a 4 appears, a ball is drawn from urn 1; otherwise, a ball is drawn from urn 2.  Urn 1 contains four red, three white, and three black balls.  Urn 2 contains six red and four white balls.

(a)    Draw a tree diagram for this experiment, and assign probabilities to the branches.  Note that the die is thrown first, and then the ball is drawn.

 

 

 

 

 

 

 

(b)   Find the probability that a red ball is drawn.

 

 

(c)    Find the probability that urn 1 was used, given that a red ball was drawn.

 

3. Before the distribution of certain statistical software every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rates for the 4 testing programs are, respectively, 0.01, 0.03, 0.02, and 0.01.

 

Draw a tree diagram:

 

 

 

 

 

 

 

 

 

 

 

(a)    What is the probability that a CD was tested and failed any test?

 

 

 

 

 

 

 

 

 

(b)   Given that a CD was tested, what is the probability that it failed program 2 or 3?

 

 

 

 

(c)    In a sample of 100, how many CDs would you expect to be rejected?

 

4. Suppose the diagram of an electrical system is given in the figure below. What is the probability that system works? Assume the components fail independently.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5. A circuit system is given in the figure below. Assume the components fail independently.

 

 

 

 

 

 

 

 

(a)    What is the probability that the entire system works?

 

 

 

 

 

 

 

(b)   Given that the system works, what is the probability that component A is not working?

 

 

 

6. Police plan to enforce speed limits by using radar traps at 4 different locations with the city limits. The radar traps at each at each of the location L₁, L₂, L₃, and L₄ are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probability of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket?