Probablity
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How many times can an event occour given all possible events. Probabilities should sum to one.
Probability
The probability of an event occouring is the number of times that the event can occour divided by all possible outcomes:
A standard (D6) die as six sides: 1, 2, 3, 4, 5, 6. What is the probability of a (fair) dice landing on 6.
number of possible outcomes: 6 number of outcomes meeting the criteria: 1 (6)
Therefore:
$$ P(6) =\frac{1}{6} $$
Inverse Probability
The probability of something not happening is 1 minus the probability of is happening:
$$ P(not X) = 1 - P(X) $$
Combining probabilities
The way in which events can be combined depends on the relationship between the events.
- If the two events cannot occour together, they are mutually exclusive. For example, rolls of a dice.
- If the two events do not affect each other, they are Indipendent, for example, rolling a dice then flipping a coin.
- If the two events interact in some way, they are not indipendent, for example, it raining and someone carrying an unbella
Mutually Exclusive
The results of mutually exclusive events can be summed (added together):
What is the probability of rolling an EVEN number on a (fair) D6:
Each side has a one in six chance (see above):
$$\begin{eqnarray} P(even) &=& P(2) + P(4) + P(6) \\\ &=& \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \\\ &=& \frac{3}{6} \\\ &=& 0.5 \end{eqnarray}$$
We can also use the ‘counting outcomes’ approach from earlier to verify this:
- number of possible outcomes: 6 (1, 2, 3, 4, 5, 6)
- number of outcomes meeting the criteria: 3 (2, 4, 6)
$$\begin{eqnarray} P(even) &=& \frac{3}{6}\\\ &=& 0.5 \end{eqnarray}$$
Independent Events
The probability that two indipendent events occours in the multiplication of those two events.
$$\begin{eqnarray} P(X \cap A) &=& P(X) \times P(A) \\\ \end{eqnarray}$$
If:
- the probability of X happening is 0.3 (P(X) = 0.3)
- the probability of A happening is 0.2 (P(A) = 0.2),
then the probability of X AND A both happening is:
$$\begin{eqnarray} P(X \cap A) &=& P(X) \times P(A) \\\ &=& 0.3 \times 0.2 \\\ &=& \frac{3}{6} \\\ &=& 0.06 \end{eqnarray}$$
Non-Indipendent
If two events are not indipendent then the probability of B occouring, given that A occoured is defined as:
$$ P(B | A) = \frac{P(A \cap B)}{P(A)} $$
Where the indipendent version of the probability is devided by the likelyhood of the first event occouring.