What is the probability that a class of 30 students out of 300 would have a final average grade higher than a 90?

95   96   94   58   67   94   89   88   97   76
50   94   61   63   78   90   79   81   98   64
86   91   92   95   85   79   89   88   95   92
94   65   40   80   93   81   95   89   91   88
96   58   88   77   98   91   97   79   93   94
92   91   61   74   90   89   80   95   96   91
80   95   90   85   77   58   98   99   100   97
95   89   85   86   71   78   88   94   72   68
97   83   88   62   89   93   42   98   89   59
88   79   95   93   80   92   96   93   89   86
96   99   93   77   59   93   89   88   92   93
88   97   80   89   93   86   95   97   93   35
68   99   94   84   58   67   93   98   88   98
88   85   84   92   80   87   95   90   93   98
93   91   89   64   96   87   91   95   96   84
89   88   97   65   97   88   79   95   92   58
75   83   95   91   92   87   89   84   89   92
60   86   87   46   91   93   95   96   98   83
93   98   83   75   97   59   93   89   47   94
86   91   92   95   88   83   87   95   88   96
93   79   85   95   78   80   81   89   97   89
89   95   94   88   68   85   83   95   84   78
89   87   98   80   93   96   88   97   82   93
43   87   54   93   95   97   93   87   94   94
92   90   95   68   83   88   93   95   89   97
93   79   87   95   97   89   96   91   98   86
86   89   56   68   91   98   64   97   96   95
89   88   85   79   95   96   94   57   99   79
95   97   96   77   75   60   87   67   63   84
95   96   91   89   93   95   69   86   61   53

Which of the following investments is riskier? Why?

For certain software, independently of other users, the probability is 0.07 that a user encounters a fault. What are the chances of the 30th user is the 5th person encountering a fault?

A queue with one server without buffer, the probability of a customer’ arrival and departure in a time unit is p and q respectively. Please try to 1) give the one step state transition probability matrix. 2) give the balance equations. 3) calculate the limiting probabilities for p=0.3 and q=0.5. (12 points)

Two teams, A and B, are playing a series of games. Assume

1. probability that A won a game is p

2. result of a game will not aect result of the next game

Find the range of p such that team A has the advantage in a best four of seven series.

Compute the relative probability of finding an electron at the bottom of the conduction band relative to the probability of finding an electron at the top of the valence band in a silicon crystal at a temperature of (a) 3 K and (b) 300 K. Use Fermi-Dirac statistics. Compare your answer with the one given by the Boltzmann equation.

Given a normal distribution with u = 70 and o= 20, what is the probability that

1. X>110
2. X<10
3. X < 70 or x> 130
4. Between what two x values (symmetrically distributed around the mean) are 70% of the values

1a.) If two events A and B are non-overlapping, then the formula for computing the probability of theirunion is: ?(?) + ?(?) . This statement is true or false?

1b.) A survey of 1,000 adults in a small town gave the following data. Round answers to the nearest tenth of a percent.

If an adult in the town is selected at random, find the probability that: a.) The adult is either female or employed.

b.) The adult is neither male nor employed.

c.) The adult is female, given the adult is unemployed.

d.) The adult is unemployed, given the adult is female.

Assuming that P equals .60 and the sample size is 1,000, what is the probability of observing a sample proportion that is at least .64.

Theory: A vector with nonnegative entries is called a probability vector if the sum of its entries is 1. A square matrix is called right stochastic matrix if its rows are probability vectors; a square matrix is called a left stochastic matrix if its columns are probability vectors; and a square matrix is called a doubly stochastic matrix if both the rows and the columns are probability vectors.

**Write a MATLAB function

function [S1,S2,P]=stochastic(A)

which accepts a square matrix A with nonnegative entries as an input. An output P will be a stochastic matrix, when it can be created according to the instructions given below. You will also calculate and output the row vectors S1=sum(A,1) and S2=transpose(sum(A,2)) in your code, where the entries of the vectors sum(A,1) and sum(A,2) are the sums of the entries of A down each column and across each row, respectively You may also employ the conditional “if” statements and logical function all.

**First, the function has to check whether a matrix A contains both a zero column and a zero row. If yes, the output has to be a message “A is not stochastic and cannot be scaled to stochastic”. (Meaning: it is neither right- nor left-stochastic and cannot be scaled to either of them.) The output P in this case will be an empty matrix, P= [ ].

**Then, the function checks whether a matrix A is: (1) doubly stochastic, (2) only left stochastic, (3) only right stochastic, or (4) neither left nor right stochastic but can be scaled to stochastic.

In the cases (1)-(3), output the corresponding messages that comment on the types of the matrices and assign to the output P the matrix A itself.

**When you are working with the case (4), output a message “neither left nor right stochastic but can be scaled to stochastic”. If the vector S1 does not have any zero entry, use this vector to modify A into a left-stochastic matrix P by scaling each of its columns by the reciprocal of the corresponding entry of S1, and output a message that you are scaling A to a left-stochastic matrix P. If S1 has a zero entry, use the vector S2 to modify matrix A into a right-stochastic matrix P by scaling each of its rows by the reciprocal of the corresponding entry of S2, and output a message that you are scaling A to a right-stochastic matrix P. Output the matrix P.

**Type the function stochastic in your Live Script.

**Run the function [S1,S2,P]=stochastic(A) on each of the matrices below (display the matrices in your Live Script file):

(a)A=[0.5, 0, 0.5; 0, 0, 1; 0.5, 0, 0.5]

(b)A = transpose(A)

(c)A=[0.5, 0, 0.5; 0, 0, 1; 0, 0, 0.5]

(d)A=transpose(A)

(e)A=[0.5, 0, 0.5; 0, 0.5, 0.5; 0.5, 0.5, 0]

(f)A=magic(3)

(g)A=diag([1,2,3])

(h)A=[0, 0, 0; 0, 0.5, 0.5; 0, 0.5, 0.5]

(k)A=randi(10,5,5);A(:,1)=0;A(1,:)=0

NOTE: Make sure that you will verify that your outputs and the messages match the corresponding definitions of stochastic matrices. If they don’t, make corrections in your code!