MA-110 Finite Math Comprehensive Exam Review

What is the meaning of A ⊆ B?

A is a subset of B

What is the meaning of A ⊄ B?

A is not a subset of B

What is the meaning of A ⊂ B?

A is a proper subset of B (has less elements that B)

What is the meaning of Ø?

The empty set

What is the meaning of a ∈ B?

a is an element within B

How do you find the odds of a given probability?

By taking the ratio of the probability of the event occurring to the probability of the event not occurring. This can be expressed as odds = P(E) / (1 - P(E)).

How do you find probability given odds?

By adding together the two elements of the ratio (7:3 = 7 + 3) and then making that number the denominator below the first element of the ratio (7/10)

How do you find standard deviation?

calculate the mean of all data points, then subtract the mean from each data point, square the results, find the average of the squared differences (called variance), and finally take the square root of the variance to get the standard deviation

What is a saddle point?

an element which is both the largest element in its column and the smallest element in its row

To multiply two matrices, the:

Columns of A must = the rows of B

The dimensions of the matrix AB will have:

The rows of A and the columns of B

One row dominates another when:

all values in the dominant row are greater than the corresponding values of the dominated row

To find the z value of a given value:

z = x (whatever value) - mean / Standard Deviation

A probability vector must be:

Only one row, with nonnegative entries that add up to 1

A transition matrix must be:

A square matrix, where all entries are between 0 and 1 and the sum of any row in the matrix is 1.

To find share of a population after a given time in a transition matrix:

multiply the initial probability vector by A^x where x is the number of repetitions

The probability of “x” successes in “n” trials is:

_nC_x · p^x · (1-p)^n-x

If the random variable x can take on the n values x1, x2, x3… w/ p1, p2, p3 respectively then expected value is:

E(x) = x₁p₁ + x₂p₂ + x₃p₃ …

The mean of a grouped distribution where f = frequency, x = midpoint, and n = Σf (Σ = sum of) is:

Σ x · f / n (multiply all midpoints by their frequencies, then divide the sum of all those values by the sum of all frequencies)

What is the simple formula for conditional probability?

P(A|B) = P(B|A) · P(A) / P(B)

How to find a missing probability in a conditional probability problem?

P(E) = P(F₁) · P(E|F₁) + P(F₂) · P(E|F₂) …

What is formula for conditional probability when a probability is missing?

P(A|B) = P(B|A) · P(A)

P(A) · P(B|A) + P(A’) · P(B|A’) …

How to find distinguishable permutations when elements are repeated? (like a word such as OPINION)

n! / (n1! n2! ... * nk!)

where n is the total number of letters and n1 and n2 and so forth are the number of the repeated letters within the word

A die is rolled ten times, find the probability of rolling no more than 4 ones:

P(no more than 4 ones) = P(0 ones) + P(a single one) + so forth