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Probability

When calculating probability, you are dealing with experiments, which are carried out randomly. The task could be to calculate the probability of obtaining one specific result out of all possible results.

This subject is closely related to combinatorics, where you count the number of combinations or permutations of experiments.

The probability of some events to happen is defined as a value between 0 and 1, where 0 is ”it never happens” and 1 is ”it always happens”.

Probability can also be given as percentages from 0% to 100%.

In order to understand the calculation of probability, you should know about set theory and combinatorics first.

Concepts of probability

Sample space

The sample space is all possible outcomes of an experiments.

For example:

When rolling a die, there are six possible outcomes. You can get 1,2,3,4,5 or 6. The sample space is denoted by the Greek letter Omega \Omega:

\Omega=\{1,2,3,4,5,6\}

Sample points

A sample point is one of the possible outcomes. All outcomes in the sample space are sample points.

Favorable outcomes

If you role a die and you want to obtain “1” or “2”, then “1” and “2” are favorable outcomes.

Symmetric probability

The probability of each outcome is the same. For example, when rolling a die the probability of obtaining “2” is the same as the probability of obtaining “6”. This is called symmetric probability.

Event

An event is one or more outcomes.

Impossible event

An impossible event is one, which will never happen. For example, it is impossible to obtain “7” when rolling an ordinary die.

The probability of an impossible event has the value of 0.

Sure event

A sure event is an event, which always happens. For example, it's a sure event to obtain a number between “1” and “6” when rolling an ordinary die.

The probability of a sure event has the value of 1.

Statistical probability

When you make a series of experiments, the outcomes of which are recorded, you can calculate the statistical probability of specific events.

The frequency of the outcomes (the number of times an outcome occurs) and the percent frequency (the percentage of each outcome’s frequency in relation to the total number of experiments) is noted in a table.

For example, a matchbox is tossed 100 times. Each time we will record, which side is facing upward, when it lands. It can either be the front side, the back side, the end faces or the striking surfaces.

FrontBackEnd facesStriking surfaces
Frequency3841813
Percent frequencyfrac{38}{100}*100\%=38\%frac{41}{100}*100\%=41\%frac{8}{100}*100\%=8\%frac{13}{100}*100\%=13\%


So for example, the statistical probability for the matchbox to land with the front facing upward is 38%.

Since experiments like this are based on coincidence, the statistical probability is not an exact probability, and you should rather consider it as a trend for something to happen. However, the more experiments you make, the more correct it will be.

The formula for symmetric probability

If all outcomes have the same probability, it is given by the formula:

P(A)=frac{favorable outcomes}{sample space}

where A is the description of the event you want to calculate the probability of.

For example, you want to calculate the probability of obtaining an even number when rolling an ordinary die.

The number of favorable outcomes is 3, because an even number is either “2”, “4” or “6”.

The sample space is 6, since the die has six sides.

The probability of obtaining an even number, when rolling the die, is then given by:

P(even)=frac{3}{6}=frac{1}{2}=0.5

So in half of the die rolles, you should obtain an even number.

Rule of addition

P(A U B)=P(A)+P(B)

The rule of addition says that the probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that even B occurs.

For example:

What is the probability of obtaining 1 or 2 when rolling an ordinary die?

P(1)=frac{1}{6}

P(2)=frac{1}{6}

P(1 or 2)=frac{1}{6}+frac{1}{6}=frac{2}{6}

The formula only applies, if the two sets of outcomes are disjoint, that is, they have no outcome in common.

If some of the outcomes in A is the same as the outcomes in B, you must use the following formula, where the union of A and B is subtracted, in order not to count their common outcomes twice.

P(A U B)=P(A)+P(B)-P(A n B)

When rolling an ordinary die, what is the probability of obtaining

“1” or “2“

or

“2” or “3”

?

P(1 or 2)=frac{2}{6}

P(2 or 3)=frac{2}{6}

P(\{1,2\} n \{2,3\})=P(2)=frac{1}{6}

The probability is:

P(\{1,2\} U \{2,3\})=P(1 or 2)+P(2 or 3)-P(2)=

frac{2}{6}+frac{2}{6}-frac{1}{6}=frac{3}{6}=frac{1}{2}

Complement – the opposite probability

Sometimes you want to calculate the probability for something not to happen.
This is called the complement, given by:

P('A)=A^C=1-P(A)

The complement is either denoted with a horizontal bar above the letter or a raised c.

For example, the probability of obtaining “6” with an ordinary die is one sixth:

P(6)=frac{1}{6}

The complement of this, which is the probability of not obtaining “6”, is given by:

P(\bar{6})=1-P(6)=1-frac{1}{6}=frac{5}{6}

So when we want to obtain “6” by rolling an ordinary die, the favorable outcome is 6:

A=\{6\}

and the complement to A is:

A^C=\{1,2,3,4,5\}

Dependent events

Two events are dependent, if they are affected by each other, that is, the probability of Event A to happen will change, if Event B happens first. This is called the conditional probability:

P(A|B)=frac{P(A n B)}{P(B)}

(If Event B has happened, what is the probability of Event A to happen?)

For example, what is the probability of obtaining “6” when rolling an ordinary die, if you know for sure that the outcome is an even number?

B: It’s and even number. B=\{2,4,6\}

A: it’s a “6”. A=\{6\}

The probability of obtaining an even number:

P(B)=frac{3}{6}

Hence the probability of obtaining “6”, when you know it’s an even number, is given by:

P(A|B)=frac{P(A n B)}{P(B)}=frac{frac{1}{6}}{frac{3}{6}}=frac{2}{6}=frac{1}{3}

Independent events

Independent events are the events, which are not affected by each other. The probability for Event A to happen has nothing to do with Event B. In that case the following rule applies:

P(A|B)=P(A)

For example, you want to calculate the probability of obtaining “6” when rolling a die, if you have obtained “heads” after having flipped a coin.

A: You will obtain “6” when rolling a die.
B: You will obtain “heads”, when flipping a coin.

Since the flip of the coin has nothing to do with the roll of the die, the probability of obtaining a “6” is still one sixth:

P(A|B)=frac{1}{6}

which is the same as:

P(A)=frac{1}{6}