Solution

Total number of ways = 6 × 6 = 36 ways.

Favorable cases = (1, 6) (6, 1) (2, 5) (5, 2) (3, 4) (4, 3)

P (A) = 6/36 = 1/6

Solution

First, calculate the total of the five test grades and the final exam (counted twice).

Total grades = 70 + 75 + 80 + 85 + 90 + 85 + 85.

Total = 570.

Divide the total by the number of tests (6, since the final counts as two).

Average = 570 / 6 = 95.

Solution

Probability of the problem getting solved = 1 – (Probability of none of them solving the problem)

Probability of problem getting solved = 1 – (5/7) x (3/7) x (5/9) = (122/147)

Solution

Solution

Solution

Solution

Solution

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Solution

There are a couple of ways of thinking about this. Firstly we can consider each bun separately. The probability that the first one has the cherry is 1/5 and the probability that the second has the cherry is (4/5)*(1/4) = 1/5 (in this case the first one has to not be a cherry bun). The probability that I have the cherry bun if therefore 2/5.

Otherwise, you can think of partitioning the 5 buns into a group of 2 and a group of 3

something like:

We then apply the cherry at random and the probability that it lands on one of the two that we have picked is 2/5.

Solution

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Solution

Let us compute the conditional probability mass function of X given Y = 2. The marginal probability mass function of Y evaluated at Y = 2 is

Solution

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Solution

Given: μ = 8.2, σ = 1, n = 70

As per the Central Limit Theorem, the sample mean is equal to the population mean.

Hence, μx = μ = 8.2 minutes

⇒ σx = 0.11 minutes

Solution

Solution

Solution

The domain of both the functions are real numbers, so there is no need to modify the domain for the first function in any case.

For f∘g(x),

Solution

Solution

This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "[" for the beginning.

Minimum: 5 inclusive, maximum: infinity

Range: [5,∞)

Solution

Begin by breaking this into steps. I will begin by computing the

g(6) step, because that will make the late steps much more manageable

Solution

Solution

= 1/3

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Solution

Observing the function we can say that the function f(x) is defined for all the values of x except for the values where, the denominator of the function is zero.

Solution

Solution

The first order partial derivatives are

So the critical points are the solutions of

Solution

Solution

Solution

Using Gauss Divergence theorem,

∴ The volume enclosed by S will be 10/3

Solution

Solution

Let us find the first-order partial derivative with respect to x and y to determine the critical points.

Now, the second equation from first and equating to zero, we get