Show that the solution to the recurrence T(n) = T(n/4)+T(3n/4) + cn, where c is a constant, is Omega of Ω( nlogn ) by using a recursion tree. Draw the recursion tree and show how you obtain this answer. To do so, you will need to consider both a lower bound to the height of the tree and the amount of work at each level of the tree. Hint: In class we examined a similar but slightly different recurrence and showed that the solution is O ( n log n ). You can follow similar logic for this recurrence, but for Ω.

Source Code for Data Structures and Algorithm Analysis in C++ (Fourth Edition)

**(10 points)**

In many of the earlier algorithms in class, such as Insertion Sort, we analyzed worst case performance. (a) What are we typically analyzing with randomized algorithms? (b) Why do we randomize the candidates in the hiring problem of the textbook before running the algorithm and analysis?

**(10 points)**

In the probability review in class, we showed an easier way to compute the Expected value (average) of the sum of two dice, by using the Linearity of Expectations. (a) Show how you would use the Linearity of Expectations property to compute the Expected value (average) of the sum of 10 fair dice. (b) Do the same, but now compute the Expected value (average) of the sum of 10 biased dice in which the probability of obtaining value 6 is 1 and the probability of all other values are 0. Show all your steps of using Expectations and Linearity of Expectations.

**(10 points)**

In the randomized hiring problem, (a) What is the probability that the first candidate is hired?

(b) what is the probability that the fifth candidate is hired? Explain your answers.

**(10 points)**

- Data-Driven Decision Making Project