## Disjoint & Non-disjoint Events, Sample Space

### Disjoint Events

**Disjoint Events**, by definition, can not happen at the same time. A synonym for this term is **mutually exclusive**.

- The outcome of a single coin toss can not be a head and a tail.
- A student can not both fail and pass a class.
- A single card drawn from a deck can not be an ace and a queen at the same time.

In probability, disjoint event A and B is expressed as:

P(A and B) = 0

In Venn diagram representation, we represent each event by the circles, if event A and B are disjoint, we end up with two circles that don't touch each other.

#### Union of disjoint events

For disjoint events A and B, the probability of A or B happening is simply the probability of A plus the probability of B.

P(A or B) = P(A) + P(B)

### Non-disjoint Events

**Non-disjoint** events, on the other hand, can happen at the same time.

- A student can get an A in Stats and A in History at the same time.

In Venn diagram representation of events A and B that are non-disjoint, we have two circles that overlap, or in other words, join, which indicates that the probability of event A and B happening at the same time is non-zero. So it is some number between 0 and 1.

P(A and B) ≠ 0

#### Union of non-disjoint events

For non-disjoint events A and B, the probability of A or B happening is the P(A) plus P(B) minus the probability of A and B happening at the same time (Avoid double counting) .

P(A or B) = P(A) + P(B) - P(A and B)

### Sample Space

A **sample space** is a collection of all possible outcomes of a trial.

A couple has two kids, the sample space for the sex of these 2 kids is:

S = {MM, FF, FM, MF} M for Male, F for Female

If you toss a fair coin twice, the sample space are:

S = {HH, TT, HT, TH} H for Head, T for Tail

### Probability Distributions

A **probability distribution** lists all possible outcomes in the sample space, and the probabilities with which they occur.

It is important to distinguish probability distribution between **discrete** and **continuous random** variables.

In the discrete case, one can easily assign a probability to each possible value. For example, when throwing a fair die, each of the six values 1 to 6 has the probability 1/6.

In the continuous random variables, when a random variable takes values from a continuum then, typically, probabilities can be nonzero only if they refer to intervals. For example in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%.

### Complementary Events

**Complementary events** are two mutually exclusive events whose probabilities add up to 1.

If a coin is tossed once, the complement of a head is tail.

one toss | head | tail |
---|---|---|

50% | 50% |

If a coin is tossed twice, the complement of outcome head-head is simply the sum of all other three possibilities, tail-tail, head-tail and tail-head.

two tosses | head-head | tail-tail | head-tail | tail-head |
---|---|---|---|---|

25% | 25% | 25% | 25% |

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