Contents
 • one choice • many choices: permutations • many choices: combinations • many choices: combinations • independent events • dependent events • permutations without repettitions • permutations with identical elements • combinations • coditional probabilities • and or events • two outcomes • total probability theorem • Bayes rule • union sets and probability • fundamental counting principle

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 Combinatorics - Probability
compound interest

compound interest

Let's suppose we borrow 1000\$. And we will have to give each mounth 100\$ back along with an interest of 12% per year. How much will we give back in 1000/100 = 10 mounth?

1. Simple interest

The borrowed 1000\$ is called the principal. Let's write P = 1000. If the (i.e. each) year interest rate is "r", then once the related year is finished, we will have to pay back
T = P + (the interest of P) = P + I = P + P x r = P(1 + r).
If we expect to pay back P in the coming "n" years, then the interest rate will be n x r and we will give back
P + nrP = P(1 + nr).

1.1. Each month

First mounth: 100\$ + 100\$ x (12% /12) = 100 ( 1 + 1%) \$ = 101 \$
Second mounth: 100 (1 + 2%) \$ = 102 \$
...
nth nouth : 100 ( 1 + n%) \$
n = 1000/100 = 10

10th mounth: 100 (1 + 10%) \$ = 110\$
The total to pay back is : 101 + 102 + ... + 110 = 10 x 100 + ( 1 + 2 + ... + 10) =
1000 + (10 x 11 /2) = 1055 \$ . Then 55\$ extra.

1.2. At the end of the period

Here the period is equal to 10 month.

10th mounth: 1000 (1 + 10%) \$ = 1100\$ Then 100\$ extra.

2. Compound interest

For each mounth and RIGHT NOW: 100\$ + 100\$ x (12% /12) = 100 ( 1 + 1%) \$ = 101 \$
First mounth: 100 (1 + 1%) \$ = 101 \$
If the seconf amount (101\$) of the second mounth is not payed NOW, it will be charged its iterest of 1% for the next month. Then, for the next mounth: 100 (1 + 1%)2 \$ = 102.01 \$
...
For the 10th mounth: 100 (1 + 1%)10

Let's write: r = 1 + 1%
The total to pay back is : 100 (r + r2 + r3 + ... + r10)
T = 100 r (1 + r + r2 + r3 + ... + r9 ) = 100 r [(1 - r10)/(1 - r)] = 100 r [(r10 - 1)/(r - 1)]

We have:
r - 1 = 1%

1. First approximation:
(r10 - 1 = (1 + 1%)10 - 1 = (1 + 10%) - 1 = 10%
Therefore:
T = 100 (1 + 1%) (10%) /1% =
100 x 1.01 x 10 = 1010 \$

2. Second approximation:
(1 + 1%)10 = 1 + 10% + 10 x 9 x 1% x 1%/2 + ...
= 1 + 10% + 45%% + ... = 1.10 + 0.0045 = 1.1045 + ...

(r10 - 1 = 0.1045 = 10.45%
T = 100 (1 + 1%) (10.45%) /1% =
100 x 1.01 x 10.45 = 1055.45 \$

3. Third approximation:
(1 + 1%)10 = 1 + 10% + 10 x 9 x 1% x 1%/2 + ... = 1 + 10% + 45%% + ...
= 1.10 + 0.0045 = 1.1045 + ...
(r10 - 1 = 0.1045 = 10.45%

T = 100 (1 + 1%) (10.45%) /1% =
100 x 1.01 x 10.45 = 1055.45 \$

4. With Calculator:
(r10 - 1 = 0.104622125 = 10.4622125%
T = 100 (1 + 1%) (10.4622125%) /1% =
100 x 1.01 x 10.4622125 = 1056.68 \$

3. nominal and effective rates

The nominal or stated rate is the unit rate: The interest rate "r" for a related period "n" is called the effective rate. Then the nominal rate is "r/n".
The interest becomes I = P x n x (r/n), and the repayment is R = P + I = P(1 + r) for the simple interest and R = P [1 + (r/n)]n for the compound interest.
The relationship between the two rates is:
P(1 + effective) = P [1 + (nominal)]n
Therefore:
effective = [1 + (nominal)]n - 1

effective rate =
(1 + nominal rate)period - 1

Annex:

1. arithmetic series:
1 = 2 + 3 + ... + n = n(n + 1)/2, indeed:
Sn = 1 + 2 + 3 + ... + n
Sn = n + n-1 + n-2 + ... +1
2Sn = (1+n) + (1+n) + ... + (1+n) , n times. Therefore:
2Sn = n (1+n). Then:
Sn = n(n+1)/2

More generally:
First term: t1 = a
Second term: t2 = a + r
Third term: t3 = a + r + r = a + 2r
...
nth term: tn = a + r + r + r + ... + r = a + (n - 1)r

The sum is: Sn = n x a + r [1 + 2 + 3 + ... + (n - 1)] =
n x a + r x (n - 1) x n/2

Sn = n [t1 + r(n - 1)/2]

Example:

For n = 10, tn = 1, and r = 1, we have:
Sn = 10 [1 + 9/2] = 5 x 11 = 55.

2. Maclaurin polynomial:
(1 + x)n =
1 + n x + n(n - 1)x2/2! + n(n - 1)(n - 2)x3/3! + ... +

3. geometric series:
(1 + r + r2 + r3 + ... + rn) = [1 - rn+1]/(1 - r) =
[rn+1 - 1]/(r - 1)
1 + r + r2 + r3 + ... + rn = (rn+1 - 1)/(r - 1)

More generally:
First factor: f1 = a
Second factor: f2 = a x r
Third factor: f3 = a x r x r = a x r2
...
nth factor: fn = a x r x r x r x ... x r = a x rn - 1

The sum is: Sn = a x r [1 + r + ... + rn - 2] =
a x r x (1 - rn - 2)/(1 - r)

Sn = f1 r (1 - rn-1)/(1-r)

Example:

For n = 10, fn = 1, and r = 2, we have:
Sn = 1 x 2 (1 - 29)/(1 - 2) = 2 (29 - 1)/(2 - 1) = 2 (29 - 1) (29 - 1) = 1022 ≈ 210 = 1024.

2. Maclaurin polynomial:
(1 + x)n =
1 + n x + n(n - 1)x2/2! + n(n - 1)(n - 2)x3/3! + ... +

Application compound interest:
p x r
p x r2
p x r3
...
p x rn

Therefore, the total is:
T = p x r (1 + r + r2 + r3 + ... + rn)
= p x r x (rn+1 - 1)/(r - 1)

Interesr rate = P r n
P: Principal, r: interest rate, n: priod

Simple interest: P → P ( 1 + n r)
Compound interest: P → P ( 1 + r)n

effective rate =
(1 + nominal rate)period - 1

arithmetic series: tn = a + (n - 1)r
Sn = n [t1 + r(n - 1)/2]

geometric series: fn = a x rn - 1
Sn = f1 r (1 - rn - 1)/(1-r)

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