Finite Math – Page 9

How Do You Find The Future Value Of An Annuity in Google Sheets?

Google Sheets has several built in functions for working with annuities. To use these functions, we’ll start with a standard Sheets worksheet.

This worksheet contains the variables used throughout Chapter 5. Values given in a problem will be entered in column B. Values calculated by Sheets will be entered in column C. We will also assume that amounts paid out are negative and amounts received are positive.

We’ll modify the worksheet shown above. This will allow us to use Sheets to calculate the different amounts in the annuity formula,

This is done using two functions in Sheets, the FV (future value) function and the PV (present value) function. Annuities have a regular payment into or out of the account. If the payment is made at the end of the compounding period, the annuity is called an ordinary annuity. Payments are made at the beginning of the compounding period for an annuity due. In Sheets, amounts that you pay out are considered negative numbers and amount you receive are positive amounts.

Find the Future Value of the Annuity

An investor deposits $500 in a simple annuity at the end of each six-month payment period. This annuity earns 10% per year, compounded semiannually.

a. Find the future value if payments are made for three years.

Solution Since the investor is paying $500 into the annuity, the payment must be entered as a negative number in Sheets.

1. Start from the basic worksheet and enter the values shown below.

2. Click in cell C6. Type =FV( in the cell. As soon as you type the parentheses, Sheets recognizes what you are trying to do.

3. Notice that the word rate is highlighted. This indicates that you need to either type in the rate or the cell reference where the rate is located. The rate for the command is actually the interest rate per period. The annual interest rate is in cell B3 and the number of periods per year is in cell B7. We need to get the interest rate per period by typing B3/B7. You can also click in cell B3, type a /, and then click the cursor in cell B7. Now type a comma.

4. The number of periods is in cell B2. Type B2 or click the cursor in that cell. To continue to the next input, type a comma.

5. The payment is located in cell B4. The value is negative since the money is paid out. Type B4 or choose cell B4 followed by a comma.

6. Since we deposit nothing into the account initially, the present value is zero. Enter B5 or select cell B5 followed by a comma and a parentheses. Press enter on the keyboard to calculate the future value.

The future value of the annuity is $3400.96. There is an optional argument for the FV function. This argument indicates whether the annuity is an ordinary annuity (use 0 for the argument) or an annuity due (use 1 for the argument).

Ordinary Annuity: FV(B3/B7,B2,B4,B5,0)
Annuity Due: FV(B3/B7,B2,B4,B5,1)

If the last argument is not supplied, the annuity is assumed to be an ordinary annuity.

b. Find the future value if payments are made for 30 years.

Solution Thirty years of semiannual payments corresponds to n = 60. We need to make this change in the FV function, and solve for future value again. Change the value in cell B2 to 60.

The future value is now $176,791.86.

How Do You Fight Spam With Bayes’ Rule?

It might surprise you to know that in 2013, 70.7% of all worldwide emails were spam. Spam emails are unsolicited email that are sent out in bulk. To combat these emails, companies utilize spam filters provided by software companies to block the spam emails from reaching the desired recipient.

One provider, SpamTitan, advertises the following data

It blocks 99.9% of all spam email.
It blocks 0.03% of all emails that are not spam.

Based on the information above, what is the probability that a delivered email is spam?

To start a problem like this, let’s identify the relevant events.

S is the event that an email is spam
S ′ is the event that an email is not spam
B is the event the an email is blocked
B′ is the event an email is not blocked

Based on these events, we want to compute the probability that an email is spam given that it is not blocked, P(S|B′).

Let’s look at a tree diagram of the situation.

Next, we’ll label the given information on the diagram.

The key here is to recognize that the data provided by the software company are conditional probabilities. Since we know that the probabilities on branches from a single point must add to 1, we can finish labeling the diagram.

The diagram is labeled nicely, but none of the probabilities match P(S|B′). The conditional probabilities on the second set of branches are all given the event S or the event S ′. To find P(S|B′), we’ll utilize Bayes’ Rule. Start with the relationship between conditional probabilities,

and solve for P(S|B′). This gives

All of the probabilities on the right side may be found from the tree diagram.

The probabilities in the numerator are located along the branch in red through S and B′.

The probability in the denominator corresponds to all branches in green that lead to B′. Since the events along each branch are disjoint, the probabilities for each branch add. This gives us

So the likelihood that an unblocked email is spam is 0.24%.

Users are typically very tolerant of getting spam that has made it through a spam filter. However, they are not very tolerant of blocked emails turning out to not be spam. This probability is P(S′|B). We can compute this probability in a similar manner:

This likelihood equates to 0.012%. This should make customers very happy since it means that there important emails will rarely be blocked by the spam filter.

How Can You Model Data With A System of Equations?

In December of 2014, Sony released the movie The Interview online after threats to theaters cancelled the debut in theaters. As originally reported in Wall Street Journal, the sales figures reported in January contained an interesting math problem appropriate for algebra students.

The following January, Sony reported sales of 31 million dollars from the sales and rentals of The Interview. They sold the movies online for 15 dollars and rented through various sites for 6 dollars. If there were 4.3 million transactions, how many of the transaction were sales of the movie and how many of the transactions were rentals?

Continue reading “How Can You Model Data With A System of Equations?”

How Can You Model Data With A System of Equations (Continued)?

In an earlier FAQ, I mentioned that there was a second strategy for solving the Sony Math Problem. Recall the basic problem:

Continue reading “How Can You Model Data With A System of Equations (Continued)?”

How Do I Find The Inverse Of A 2 x 2 Matrix?

Suppose the given square matrix is called A. To find the inverse of any matrix, we write the matrix in a larger matrix along side an identity matrix of the same size,

$latex \displaystyle \left[ \left. A\, \right|\,I \right]$

Now use row operations to rewrite this matrix so that the identity appears on the left side. The inverse of the original matrix will be on the right side of the transformed matrix,

$latex \displaystyle \left[ \left. I\, \right|\,{{A}^{-1}} \right]$

Continue reading “How Do I Find The Inverse Of A 2 x 2 Matrix?”