MATH 1324 - Exam 1
Finding PMT (Payment per Period)
(dividing total payment by all annual payments in years) = PMT
Chapter 8.4
- additional rule for probability - complement rule - odds in favor - ratio odds
Chapter 5.2
- annual interest - compound interest - present & future value for compound interest - continuous compounding - zero coupon bond - annual percentage yield (APY)
Chapter 5.3
- annuity - ordinary annuities - payment period - sinking fund - annuities due
Chapter 9.4
- binomial distribution - bernoulli trials - binomial experiments - binomial probability - binomial distribution expected value
Chapter 8.5
- conditional probability - product rule of probability - independent/dependent events
Chapter 8.3
- experiments - sample spaces - trials - events - probability
Chapter 5.1
- interest terms - simple interest - time rules - present value for simple interest - future value for simple interest - loans - discount formula - t bill - actual rate
Chapter 9.2
- multiplication principle - factorial notation - permutations - combinations
Chapter 5.4
- payout annuities - amortization payment - remaining balance - extra payment - present value for annuities due
Chapter 9.3
- probability at random selection - application of counting; ways to choose & combinations - probability rules reclarified
Chapter 9.1
- probability distribution - histrograms - expected values
Chapter 8.1
- sets - subsets - complements
Binomial Experiment
Binomial Experiment: - same experiment repeated fixed number of times - only 2 possible outcomes; success and failure - probability of success for each trial is constant - repeated trials are independent
Binomial Distribution
Binomial distribution occurs when the same experiment is repeated many times and each repetition is independent of previous ones. - 1 outcome = success - any other outcome = failure
Properties of Probability (Formula)
Probabilities will always be between (and including) 0 and 1. A probability of 0 means that the event is impossible. A probability of 1 means an event is guaranteed to happen. Let S be a sample space consisting of n distinct outcomes suchterm-24 as s1,s2...sn An acceptable probability assignment consists of 1. Probability of each outcome is a number between 0 and 1 2. The sum of probabilities of all possible outcomes is 1 This is only for questions that look like the picture, but in a sense you could say all events are between 0 and 1, as there is either a 0% chance, 100% chance, or in between.
Remaining Balance
REMAINING BALANCE: If n payments are needed to amortize a loan and x payments HAVE been made, the remaining payments form an annuity of n - x.. So.. B (Balance) = PMT • [ 1 - ( 1 + i)^-(n-x) / i]
Solve for Monthly Withdrawals (Formula)
Rearrange the PV/FV formulas for ordinary annuities to solve for A (the monthly withdrawal amount): A = Future Value • i / [1 - (1 + i)^(-n)]
Simple Discount Loan
SIMPLE DISCOUNT LOAN: - interest is deducted in advance from the amount of the loan - balance given to borrower - full value of the loan must be paid back at maturity Most common form of simple discount loans are U.S. treasury bills (T-bills) which are short-terms loans to the U.S. government by investors. T-bills are sold at a discount from their face value and the treasury pays back the face value at maturity. As such, use the T-bill formula for any question mentioning short-term loans.
Sinking Fund
SINKING FUND: A sinking fund is a fund set up to receive periodic payments. If the payments are equal and are made at the end of regular periods, they form an ordinary annuity.
Sets
Sets are like a well-defined collection of objects. - must be clearly defined - sets are usually numbers - usually defined by capital letters - can have NO elements, called an EMPTY SET - 0 does not count as a natural number - every set is a subset of itself
Solving for Interest Rate / Time (Formula)
Sometimes, you will need to solve for interest rate or the simple interest rate's time. Here is how you can do that simple interest rate (r) = 12((final amount / principle investment) - 1) simple interest rate (t) = interest difference/(principal amount*rate)
Multiplication Principle (Formula)
Suppose N choices must be made with M1 ways to make choice 1 and M2 ways to make choice 2, Mn ways to make choice N, etc.. Then there are m1 • m2 ......... mn different ways to make the entire sequence of choices. Example: Pizza House offers 4 different salads, 3 different kinds of pizza, and 3 different desserts. How many different three course meals can be ordered? 4 • 3 • 3 = 36
T-Bill (Formula)
T-BILL = FV - D where FV is Future value and D is Discount
Term of the Annuity
TERM OF THE ANNUITY: The term of the annuity is the time from the beginning of the first payment to the end of the last period.
Zero-Coupon Bond
Unlike others bonds, with a zero-coupon bond, there are no interest payments during the life of the bond. The investor receives a SINGLE payment when the bond matures. - consists of original investment & interest compounded semiannually that it has earned.
Basic Probability Rules (Formula)
Used for Venn Diagram questions, P(A') = 1 - P(A) Hence, if in a venn diagram A has a value of 0.30, then its complement is 1 - 0.30 = 0.70 Other rules: P(A∪B) = P(A) + P(B) - P(A ∩ B) P(A∩B) = P(A) + P(B) - P(A ∪ B) P(A ∩ B') = P(A) - (A ∩ B) etc..
Conditional Probabilities (Formula)
We need to use conditional probabilities when we want to examine data further. This includes additional info. P(A|B) = Probability of A, given B (additional info) P(A|B) = n(A ∩ B) / n(B) = The probability that both A and B occur. 'n' represents the amount in each probability. This formula must be used in questions that have word like "what's the probability that x occurs, GIVEN that..." If it wants to know "what's the probability that A occurs AND that B occurs" you will need to use the formula P(A&B) = n(A ∩ B) / n(S)
Fundamental Principle of Counting
You can find the total amount of outcomes for a sample space by multiplication. Example: Rolling two pair of dice A die has 6 faces, so they each have 6 faces. 6 • 6 = 36 total possible variations in outcomes
Odds in Favor (Formula)
the ratio of the number of favorable outcomes to the number of unfavorable outcomes Odds in favor of rolling a 4 on a 6 sided die? 6 - 1 because 5 are not of our desired 4. 1 / 5 = 1:5 ratio
Subset
two types of subset: - normal subset (just called subset) - proper subset every subset is a subset of itself
∈
∈ means BELONGS to the set example: 4 ∈ {3, 4, 5} means is 4 in the set? yes it is, so it's true
∉
∉ means DOES NOT belong to the set. 3 ∉ {4, 5, 6} = true because 3 isn't there
Additional rule for probability (Formula)
For any events E and F from a sample space S Useful for needing to solve for E ∪ F P(E ∪ F) = [P(E) + P(F)] - P(E ∩ F) In other words, P(E ∩ F) cannot be larger than P(E) + P(F). The largest P(E ∩ F) can be equal to is the minimum of P(E) and P(F). Essentially, if you're trying to find the probability of event E OR event F happening, combine their possible outcomes and use the P(E) probability formula to solve. Example: There are 45 fruits, 10 are Apples, 5 are Cucumbers. Chance that an apple or a cucumber is chosen = 10+5 = 15 / 45, simplify to 1 / 3 - FOR DISJOINT EVENTS E AND F: P(E ∪ F) = P(E) + P(F)
Product rule of Probability (formula)
For any two events and B, the product rule states: dependent events: P(A ∩ B) = P(A|B) • P(B) independent events: P(A ∩ B) = P(A) • P(B)
Interest Terms
INTEREST: Interest is the fee used to pay someone else's money. PRINCIPAL: The principal is the amount of interest borrowed or deposited. T-BILL: A T-Bill is the short-term debt obligation backed by the U.S. Department of the Treasury with a one-year maturity or less.
Time Rules
If T is represented in months, the interest rate should be divided by 12. and etc. Example: T = 12 months I = 5% then 0.05 / 12 will be the interest rate when using it in formulas For days, 365 For semi-annual 2, quarterly 4, etc. For years, there is no need to divide, unless it's 1 year and 6 months, which then you will divide by 18.
Binomial Probability (Formula)
If p is the probability of success in a single trial, n is the sample size, and x is the desired sample size, the probability of x successes and n - x failures in n independent repeated trials of the experiment is: nCx • [p^x • (1 - p)^(n - x)] Remember, n C x part can and should be computed via a calculator.
Ways to choose (Formula)
In questions such as "How many ways to choose... etc" Combinations can always be used. How many ways can 4 tiles be selected from 18 tiles? 18 C 4 = 3,060 ways.
Bernoulli Trials
Individual trials, such as rolling a die once, is called a bernoulli trial. Or, bernoulli processes.
Infinite sets
Infinite sets are sets that can go on forever, indicated by " ..." so natural numbers could be {1, 2, 3, 4, ...} as the amount of natural numbers are effectively infinite. Example: {x|x is an integer} There are infinite amount of integers, so it would go on forever.
Simple Interest (Formula)
Interest on loans of a year or less is frequently calculated as simple interest. - paid only on amount borrowed or invested - not paid on past interest The simple interest I on PV dollars at a rate of interest r per year for t years is.. I = PV • (r • t) You can also calculate via: I = FV(Future Value) - PV(Present Value) Remember, this is simply to calculate interest, not the interest RATE.
Basic Probability Principle (Formula)
Letting S be a sample space of equally likely outcomes, E a subset of S.. then the probability that E occurs is: where P is probability and E is the event, P(E) = Number of favorable outcomes for E / Total number of outcomes in S Example: S = There are 20 students in a class. 4 are girls. E = The selected student is a girl Answer: 4 / 20 (simplify into 1 / 5) answer is 1 / 5 which is also 20% of a chance In questions that ask for 4 decimal points, the same rule applies, but do not treat it as a fraction, just divide the event's outcomes by the sample space and round.
Ordinary Annuities (Formula)
ORDINARY ANNUITIES: - payments made at end of each period - frequency of payments is the same as the frequency of compounding the interest where n is the period, i is interest rate FV = PMT * [ ((1 + i)^n) - 1) / i ] PV = PMT * [1 - ((1 + i)^-n) / i]
Empirical Probability (Formula)
Often used in trials/experiments questions. Simply take the needed outcome and divide it by the total sample size. Example: 100k single parents are interviewed, 39k are in the army The probability that a random single parent is chosen and is in the army is 39 / 100 = 0.39 or 39%
Payment Period
PAYMENT PERIOD: A payment period is the time between payments in an annuity.
Solving for payments per month (Formula)
PAYMENTS PER MONTH = FV / PV
Payout Annuities (Formula)
PAYOUT ANNUITIES: Payout annuities are stuff like jackpots, lottery, etc. They begin with an amount of money and make regular payments out of the account each period until the value decreases to 0. Interest = r / m where r is the annual interest rate and m is periods per year
Present Value (Simple Interest) (Formula)
PRESENT VALUE FOR SIMPLE INTEREST: The present value PV of a future amount of FV dollars at a simple interest rate r for t years is PV = FV / 1 + (r•t)
Combinations
A combination is a subset of items selected WITHOUT REGARD TO ORDER. COMBINATIONS: each choice or subset of r object gives 1 combinantion, order doesn't matter, groups, committees, sets, samples, etc. (with questions with these traits use COMBINATION not PERMUTATION! to solve) - use a calculator for this!
Probability Rule (Formula)
As a reminder, to find the probability something will not happen from an existing probability, you can do 1 - P(E) (probability an event happens)
Complements
A complement is something that complements another set. If A was all female students in a class, and U was all students in the class, then the COMPLEMENT to set A would be all the male students in the class. Essentially, the opposite. Example: {x|x is all numbers between 5 and 10} Complement: {x|x is all numbers below 5 and above 10} Complements are written with the ' next to the set name. For example, the complement of set A would be written as A' or A "prime" Hence, B = {all female students} B' - {all male students} A complement of a set can also just be the elements in U that are not in that set. Complement of an empty set would be the Universal set and vice versa.
Permutations
A permutation of a set of elements is an ordering of the elements. For instance, there are 6 permutations (orderings) of the letters A, B, and C, so like ABC, ACB, BAC, BCA, etc.. - the number of permutations of an n element set is n! PERMUTATIONS = different orderings / arrangements, schedule, etc. ORDER MATTERS (so in questions with these traits, use permutation, not combination!) - use a calculator for this
Probability Distribution
A probability distribution is a table that lists all the outcomes with the corresponding probabilities. - depends on the idea of a random variable - expected value of a probability distribution is a type of average - sum of probabilities must always equal 1 A probability distribution is invalid if any of the probabilities are below 0, above 1, or if the probabilities do not all sum up to 1.
Proper subset
A proper subset is when for example the B set contains all elements of the A set but the B set has additional elements not in the A set. A PROPER subset is written A ⊂ B Proper subsets are still a (normal) subset, just a special kind. If sets are equal and you are looking for a subset, use ⊆, but if the second set contains additional elements you will use ⊂ to denote it is a proper subset.
Set Builder Notation
A set builder notation describes a set by a common property of its elements RATHER than by a list of the elements. {x|x is an integer strictly between 3 and 7} = {4, 5, 6} A complement of this would be {x|x is an integer below 3 or below 7}. Essentially everything that the first problem was not.
(normal) Subset
A subset is a set that is part of a larger set A and B being lists, A ⊆ B would mean that every element of A is also in B A={3,2,1} B= {3,2,1} then A ⊆ B = true However, if B did not contain ALL of A then it would be A ⊈ B And if B contained additional elements not in A, it would be A ⊂ B Also, a possible subset of every set can be a ∅ (empty). As in, ∅ is a part of every set and subset. (do not count it as an element, though)
Amortization Payment (Formula)
AMORTIZATION PAYMENT: A loan is amortized is both the principal and interest are paid by a sequence of equal periodic payments. The periodic payment needed to amortize a loan may be found by A loan of PV dollars at interest rate i per period may be amortized in n equal periodic payments of PMT dollars made at the end of each period. finding monthly payment amortization/borrowing PMT = PV • (i / 1 - (1 + i)^-n)
Annual Interest
ANNUAL INTEREST: With annual simple interest you earn interest each year on your original investment.
Annuities Due (Formula)
ANNUITIES DUE: Annuities due are annuities where payments are made at the BEGINNING of each period. FV = PMT • [((1 + i)^n+1) - 1 / i] - PMT Note if the payments are made at the END of a period, it's still the same formula but we do not add + 1 to the power of the n in this equation FV = PMT * [((1 + i)^n) - 1 / i] - PMT PV = PMT / (((1+i)^n) - 1 / i) * (1+i)
Annuity
ANNUITY: A sequence of equal payments made at equal periods of time is called an annuity. - can be used to accumulate funds - future value of compounded annuity = final sum on deposit, total of all deposits and interest earned by them
Empty Set
An empty set is a set with no members, represented by ∅ symbol. - {0} represents a set with ONE element - ∅ represents an empty set completely - empty set is a subset of EVERY set
Expected Value (Formula)
An expected value, in general, is the value that is most likely the result of the next repeated trial of a statistical experiment. The probability of all possible outcomes is factored into the calculations for expected value in order to determine the expected outcome in a random trial of an experiment. where p is a probability and x is the number assigned to that probability and the numbers associated such as x•1*, x•2 • ... is representative of their own spots within a table. Therefore, Expected Value E(x) = x1•p1 + x2•p2 + x3•p3 + ... + As many problems as there are in the table, you multiply the x value assigned to each probability, then add them to the next probability pair, until you get your answer.
Intersections
An intersection is the set of all elements belonging to both of the two sets. A & B is written as A ∩ B {1,2,3,5} ∩ {1,2,3,4} = {1,2,3}, the elements A and B share. In cases of A' ∩ B' then that = all the elements NOT in sets A & B Also, {x|x is a teenager} ∩ {x|x is a senior citizen} will equal ∅ because the two sets would never have identical elements and ∅ is the symbol of an empty set When finding the intersection of two fractions, simply multiply them.
Compound Interest (Formula)
COMPOUND INTEREST: Unlike annual interest, you earn interest both on your original investment AND any previously earned interest. If PV dollars are invested at interest rate i per perioud, then the compound amount (future value) FV after n compounding periods is. A is the amount of money accumulated after n years, including interest. FV = PV • ((1+ i)^n) Remember, if it's not in years but in days, semi-annual, etc, adjust the formula. For example, for months, adjust the formula to include the 12 month calendar FV = PV • ((1+ (i / 12))^(n * 12) For PV... Same rules above apply for if months/days are used instead of years. The only difference in this formula is that we divide Future value by the equation instead of multiplying Present Value PV = FV / ((1 + i)^n)
Continuous Compounding (Formula)
CONTINUOUS COMPOUNDING: Continuous compounding is compounding more and more frequently, used in certain financial situations. Again, rules apply for if the time is in days/months instead of years. So you'd adjust it by dividing the rate by 12/365/2/4, etc etc as needed, before all else. FV = PV • (e^r•t) PV = FV / (e^r•t)
That's The End
Congrats, that's the end, wrap it up, go practice some assignments, and come again tomorrow and repeat! Go study the formula flashcards individually, too.
Discount Formula (Formula)
DISCOUNT FORMULA: D = FV * (r * n / t) where n is the maturity date and t is time and r is the interest rate and FV is Future Value
Disjoint Events
Disjoint events are 2 events that cannot both occur at the same time, like getting both a head and a tail on the same coin toss. Also called mutually exclusive events. - events are disjoint events if E ∩ F (Example events) = ∅ Example: A randomly selected person is to become a nurse E = The person is a little girl F = The person is above the age of 20 As they would have nothing in common the E ∩ F set = ∅ All events are disjoint events with themselves. As in, the complement of any event is disjoint with itself.
Expected value for Binomial Distribution (Formula)
EXPECTED VALUE FOR A BINOMIAL DISTRIBUTION: When an experiment meets the four conditions of a binomial experiment (- same experiment repeated fixed number of times - only 2 possible outcomes; success and failure - probability of success for each trial is constant - repeated trials are independent) with n fixed trials and constant probability of success p, the expected value is E(x) = n•p
Extra Payments
EXTRA PAYMENTS: Many financial planners recommend making extra payments toward the principal of the loan each month. One way to make an extra payment each year is by adding 1/12 of the regular monthly payment to EACH monthly payment. This extra amount is then applied to the principal and reduces the amount of future interest payments (1/12) • payment required each month
Future Value (Simple Interest) (Formula)
FUTURE VALUE (OR MATURITY VALUE) FOR SIMPLE INTEREST: The future value FV of PV dollars for t years at interest rate r per year: FV = PV(1 + (r •t))
Factorial Notations (Formula)
Factorial Notations.. Instead of writing 5•4•3•2•1, we write 5! (5 factorial), if n = 1 then 1! = 1, and if n = 5, then 5! = 120 Basically, multiply every positive integer LESS than the number!
Probability combination
Take a problem such as this: A manufacturing company performs a quality-control analysis on the ceramic tile it produces. Suppose a batch of 26 tiles has 9 defective tiles. If 7 tiles are sampled at random, what is the probability that exactly 1 of the sampled tiles is defective? To solve, find the amount of combinations from 26 C 7 = 657,800. Also known as the amount of ways to select 7 tiles from a batch of 26. Find the amount of ways to choose 1 defective tile from the 9 tiles. So 9 C 1 = 9. Find the amount of ways to choose 6 good tiles (since 1 of the 7 is defective) from the 17 good tiles of the batch of 26. 17 C 6 = 12,376 Using the multiplication principle, 9 * 12,376 = 111,384 ways to choose exactly 1 defectile tile Now to find the answer 111,384 / 657800 = 0.1693 All problems like this must be solved in a similar way as this is the best explanation I could find for solving problems like this in relation to the chapter. Additionally, for questions saying "at most", for example, "At most 3", you must use the equation above for 0, 1, and 2. Then add your answers together to get the final answer. For questions that say "at least", for example "At least 3" you must do the above for 0, 1, 2, and add the answers together. Then, minus one from your final answer to get the real answer. Weird rule but it is what it is.
Annual Percentage Yield (APY) (Formula)
The APY r E is the simple interest rate needed to produce the same amount of interest in one year as the nominal rate does with more frequent compounding over one year. APY = (1 + (r/n))^n) - 1 where r is rate and n is times per year
Histograms
The information in a probability distribution is often displayed graphically as a special kind of bar graph. The bars of a histogram all have the same width, usually 1 unit. The heights of the bars are determined by the probabilities. Pay attention closely to the bars and their values to answer questions properly.
Sample Space
The sample space for a random experiment is the set of ALL possible outcomes. Example: A day of the week is chosen for a holiday. S (Sample Set) would equal = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
Union
The set of all elements belonging to two sets (in this case A & B) is called the UNION of the two sets, presented as: A ∪ B A or B = A ∪ B so... {1,2,3,4} ∪ {1,2,3,6} is then equal to {1,2,3,4,6}
Universal Set
The universal set in a particular discussion is a set that contains all of the objects BEING discussed.
Finding Subsets
There are two possibilities to each element of a set. Therefore, you can find the number of subsets in a set by using the equation: (^ = to the power of) 2^E where E is the number of elements in the set.
Dependent/Independent Events
To find if two events are independent/dependent on each other, given events F and E, if P(F|E) = P(F), then it is independent, if not, then it is dependent. Sometimes it will just be clear, for example: A person obtaining a PHD then getting a position as a professor. That is clearly a DEPENDENT event, but A person buying ice cream and listening to pop music Are not dependent, it is independent because they are irrelevant to each other.
Equal Sets
Two sets are equal if they contain exactly the same elements. REGARDLESS of order {5, 6, 7} = {7, 6, 5} is true
Disjoint Sets
Two sets that have NO elements in common are called DISJOINT SETS. So, {1,2,3} ∩ {4,5,6} = ∅ If A n B = ∅, they share no elements
