Suppose 100 people all toss a hat into a box and then proceed to randomly

pick out of a hat. What is the expected number of people to get their own hat back.

Suppose you pick two cards from a deck of 52 playing cards. What is the

probability that they are both queens (Pick the closest answer)?

Let A be the matrix given by A = [ 2  -1  1 

                                                        1   2   -1

                                                         1   -1   2]

Is the matrix given above diagonalizable?

Let A be the 3 x 3 matrix as follows

A = [ 8 -6 2

        -6  7 -4

         2 -4 3]

What is sum of eigenvalues of A ?

Let 𝑋 be a random variable with range [0, 1] and cdf

𝐹(𝑋) = 2𝑥^2 − 𝑥^4 for 0 ≤ 𝑥 ≤ 1.  Compute 

𝑃 ( 1/4 ≤ 𝑋 ≤ 3/4 )

Suppose that 𝑃(𝐴) = 0.4, 𝑃(𝐵) = 0.3 and 𝑃((𝐴 ∪ 𝐵)) = 0.58. Are 𝐴 and

𝐵 independent?

Suppose that there are ten students in a classroom. What is the probability

that no two of them have a birthday in the same month (Pick the closest answer)?

If A is 3 x3 matrix with all real entries, has an eigenvalue 2i , then its other two eigenvalues

can be

A continuous random variable X follows a normal distribution with mean μ and standard deviation σ. The normal curve has specific geometric properties, including points where the concavity of the curve changes. What is the exact length of the interval over which the normal probability density function is concave downward?

Suppose that 𝑋 takes values between 0 and 1 and has probability density

function 2𝑥. Compute Var(𝑋^2).