Q: A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

The probability is $$\frac{5!}{3!2!} \times \left(\frac{1}{2}\$$right)^5$$ = \frac{10}{32} = \frac{5}{16}.$$

Question 1

Which of the following best describes the difference between a permutation and a combination?

Accepted Answer

A permutation is an arrangement of objects where order matters, while a combination is an arrangement where order does not matter.

Question 2

Suppose a medical researcher wants to select 3 patients from a group of 10 for a clinical trial. How many different groups of 3 patients can be formed?

Accepted Answer

Using the formula $$\binom{n}{r}$$, the answer is $$\binom{10}{3} = 120.$$

Question 3

A fair coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Accepted Answer

The probability is $$\frac{5!}{3!2!} 	imes \left(\frac{1}{2}\$$right)^5$$ = \frac{10}{32} = \frac{5}{16}.$$

Question 4

Which of the following statements about conditional probability is correct?

Accepted Answer

Conditional probability measures the likelihood of an event occurring given that another event has already occurred.

Question 5

A bag contains 4 red, 3 blue, and 2 green marbles. If one marble is drawn at random, what is the probability it is blue?

Accepted Answer

The probability is $$\frac{3}{9} = \frac{1}{3}.$$

Question 6

Which formula is used to calculate the expected value $$E of a $$discrete random variable $$X$$ with possible values $$x_1$$, $$x_2$$, ..., x_n$$ and corresponding probabilities p_1$, p_2$, ..., p_n$$?

Accepted Answer

$$E = \sum_{i=1}^{n} x_i p_i$$

Question 7

A normal distribution is characterized by which two parameters?

Accepted Answer

Mean ($$\mu) $$and standard deviation ($$\sigma)$$

Question 8

Which of the following is the correct formula for Bayes' theorem?

Accepted Answer

$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

Question 9

A system of equations is given by $$\begin{cases} 2x + 3y = 8 \ x - y = 2 \end{cases}. $$What is the solution $$(x, y)$$?

Accepted Answer

Question 10

Which matrix operation is used to solve a system of linear equations represented as $$AX = B$$?

Accepted Answer

Multiplying both sides by the inverse of $$A$$, $$A^{-1}$$, so $$X = A$$^{-1}$$B$$

Question 11

In a Markov chain, what does the transition matrix represent?

Accepted Answer

The probabilities of moving from one state to another in one step.

Question 12

A company produces two products, A and B. The profit for each is given by $$P = 5x + 3y$$, where $$x$$ and $$y$$ are the number of units produced. If the company can produce at most 100 units in total, which inequality represents this constraint?

Accepted Answer

$$x + y \leq 100$$

Question 13

Which of the following is NOT a property of the normal distribution?

Accepted Answer

The area under the curve is infinite.

Question 14

If a set $$A$$ has 6 elements and set $$B$$ has 4 elements, how many elements are in $$A \cup B if$$ $$A$$ and $$B$$ are disjoint?

Accepted Answer

10

Question 15

Which of the following best describes the purpose of linear programming in real-world applications?

Accepted Answer

To maximize or minimize a linear objective function subject to linear constraints.

Question 16

A Venn diagram is most useful for illustrating which mathematical concept?

Accepted Answer

Relationships between sets, such as intersections and unions.

Question 17

If the probability of event $$A is$$ $$0.4$$ and the probability of event $$B is$$ $$0.5$$, and $$A$$ and $$B$$ are independent, what is the probability that both $$A$$ and $$B$$ occur?

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 18

Which of the following is the formula for the standard deviation $$\sigma of a $$discrete random variable $$X$$ with mean $$\mu$$ and possible values $$x_1$$, $$x_2$$, ..., x_n$$ with probabilities p_1$, p_2$, ..., p_n$$?

Accepted Answer

$$\sigma = \sqrt{\sum_{i=1}^{n} (x_i - \mu)^2$ p_i}$$

MATH-M 118: Finite Mathematics Syllabus and Course Overview

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