Formulae - Statistics

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Venn Diagrams

General Rules:

P(A) = probability of even 'A' occurring

P(A`) = probability of 'A compliment' (NOT 'A') occurring = 1 - P(A)

P(AnB) = probability of both 'A' AND 'B' occurring (the intersection)

P(AuB) = probability of either 'A' or 'B' occurring (the union)

Golden Rule:

P(AuB) = P(A) + P(B) - P(AnB)

Special Cases:

Mutually exclusive:

If the events of 'A' and 'B' are separate, they are said to be 'mutually exclusive'

∴ there is no intersection, P(AnB) = 0

Independent Events:

If the events of 'A' and 'B' are independent, they are said to be 'independent events'. This is in the case of flipping a coin and rolling a die; the outcome of one has no bearing on the other.

In this case P(AnB) = P(A) * P(B)

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Binomial Distribution

Bernoulli Experiment:

An experiment with only two possible outcomes. These could include:

- True or False

- Heads or Tails

- 4 or not 4 (on a die)

Pascal's Triangle:

This triangle dictates the number of combinations for every possible outcome in a Bernoulli experiment with any given number of trials. An example using the 4th row (1, 3, 3, 1) can be seen below:

"A coin is flipped 3 times:

No. of heads: 0 1 2 3

Combinations: 1 3 3 1 "

Other facts:

- An unknown number in the row below can be found by adding the two numbers above it if they are known (1 + 2 = 3, as seen in rows 3 & 4)

- Each row summates to a power of two (2^0 = 1, 2^1 = 1 + 1 = 2, 2^2 = 1 + 2 + 1 = 4, e.t.c)

- Adding each number in a diagonal line will result in a term from the Fibonacci sequence

Generating the Triangle:

Generating terms from the triangle can be done with the simple expression "nCr", where:

"n" is the number of trials;

"r" is the number of successes required

So, nCr means the number of combinations possible when we want r successes from n trials!

The formula for calculating nCr is as follows:

nCr = n! / r! * (n - r)!

∴ in the case n = 6, r = 4,

nCr = 6! / 4! * (6 - 4)! = 6! / 4! * 2!

= 6*5*4*3*2*1 / (4*3*2*1) * (2*1)

= 6*5 / 2*1 = 30 / 2 = 15

(Remeber: n! is a factor of [n + 1]!)

Using the triangle to find the probability of an event:

In these cases, we define x as the number of successes the trial achieves. We say that x is binomial with n trials and p probability of success, or XnB(n, p)

The probability of an event occurring can be found with nCr using the formula:

P(x = r) = nCr * p^r * q^(n - r), where:

"n" is the number of trials, "r" is the number of required successes;

"p" is the probability of success, and "q" is the probability of failure

This formula can then be used in the cases of P(x = r) and P(x >=/<= r) to find solutions to a binomial problem

Finding the probability of P(x <= r) with the equation:

In order to find the probability of x <= r, we simply repeat "P(x = r) = nCr * p^r * q^(n - r)" for every value that r can be in correspondence to x.

Finding the probability of P(x <= r) with the table:

The binomial distribution table shown gives the probability of every expression of x <= r for most values of n. This is used as a quicker alternative to using the formula.

∴ in the case XnB(20, 0.4),

P(x <= 9) = 0.7553 = 20C0 * 0.4^0 * 0.6^20 + ... + 20C9 * 0.4^9 * 0.6^11

Other notes concerning the table:

P(x < r):

P(x < r) is not available on the tables ∴ P(x <= r - 1) should be found instead when asked in the format

Probabilities greater than 0.5:

Probabilities greater than 0.5 are not given on the tables. This means that a variable "y" must be introduced, where:

"y" is the number of failures

This is stated as YnB(n, q) because the probability of x is less than or equal to r is the same as the probability of y is greater than or equal to r + 1.

∴ P(x <= r) = P(y >= r + 1)

P(x >= r):

The table only shows values for P(x <= r) so we must change P(x >= r) to 1 - P(x <= r) in order to find its probability (this applies to y as well).

P(a <= x <= b):

To find the probability of P(a <= x <= b) we must treat the two inequalities as separate: we have P(x >= a) and P(x <= b);

Then, we convert P(x >= a) to 1 - P(x <= a - 1) so that we can find it in the table;

Finally, because we want the probability of P(a <= x <= b) and not P(a <= x <= n), we do P(x <= b) - P(x <= a - 1). This can be seen with the lists below:

If a = 5, b = 10 & n = 15 then P(x <= b) - P(x <= a - 1) will be the probability of x falling in the bold section:

1 2 3 4 | 5 6 7 8 9 10 | 11 12 13 14 15

c d

"c" represents P(x <= a - 1);

"d" represents P(x <= b)

However, if we wrote 1 - P(x <= a - 1) we would get this:

1 2 3 4 | 5 6 7 8 9 10 11 12 13 14 15 |

c d

"c" represents P(x <= a - 1);

"d" represents P(x = n) = 1

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Poisson Distribution

Definition:

This is a special case of a Bernoulli experiment where the number of successes is, in theory, infinite. For this type of trial, we use the mean number of successes to find a probability. This is explained in the example below:

"A coin is flipped 10 times, meaning a maximum of 10 successes can be achieved (this is Binomial). However, if the number of red cars that pass by a building is recorded for an hour, the number of possible successes are effectively infinite (this is Poisson)."

Formula:

Poisson is defined as XnPo(λ), with λ as the mean value;

The Poisson formula is:

P(x=r) = e^-λ * λ^r / r!

Table:

The probability of an inequality (x < 5) can be found using a table, the same as with Binomial. The tables are < tables.

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