*Events can be Independent, Mutually Exclusive or Conditional !*

Life is full of random events!

You need to get a “feel” for them to be a smart and successful person.

The toss of a coin, throw of a dice and lottery draws are all examples of random events.

## Events

When we say “Event” we mean one (or more) outcomes.

Example Events:

- Getting a Tail when tossing a coin is an event
- Rolling a “5” is an event.

An event can include several outcomes:

- Choosing a “King” from a deck of cards (any of the 4 Kings)
**is also**an event - Rolling an “even number” (2, 4 or 6) is an event

## Independent Events

Events can be “Independent”, meaning each event is **not affected** by any other events.

This is an important idea! A coin does not “know” that it came up heads before … each toss of a coin is a perfect isolated thing.

Example: You toss a coin three times and it comes up “Heads” each time … what is the chance that the next toss will also be a “Head”?

The chance is simply 1/2, or 50%, just like ANY OTHER toss of the coin.

What it did in the past will not affect the current toss!

Some people think “it is overdue for a Tail”, but *really truly* the next toss of the coin is totally independent of any previous tosses.

Saying “a Tail is due”, or “just one more go, my luck is due” is called **The Gambler’s Fallacy**

(Learn more at Independent Events.)

## Dependent Events

But some events can be “dependent” … which means they **can be affected by previous events** …

### Example: Drawing 2 Cards from a Deck

After taking one card from the deck there are **less cards** available, so the probabilities change!

Let’s say you are interested in the chances of getting a King.

For the 1st card the chance of drawing a King is 4 out of 52

But for the 2nd card:

- If the 1st card was a King, then the 2nd card is
**less**likely to be a King, as only 3 of the 51 cards left are Kings. - If the 1st card was
**not**a King, then the 2nd card is slightly**more**likely to be a King, as 4 of the 51 cards left are King.

This is because you are **removing cards** from the deck.

Replacement: When you put each card **back** after drawing it the chances don’t change, as the events are independent.

Without Replacement: The chances will change, and the events are **dependent**.

You can learn more about this at Dependent Events: Conditional Probability

## Tree Diagrams

When you have Dependent Events it helps to make a “Tree Diagram”

### Example: Soccer Game

You are off to soccer, and love being the Goalkeeper, but that depends who is the Coach today:

- with Coach Sam your probability of being Goalkeeper is
**0.5** - with Coach Alex your probability of being Goalkeeper is
**0.3**

Sam is Coach more often … about 6 of every 10 games (a probability of **0.6**).

Let’s build the Tree Diagram!

Start with the Coaches. We know 0.6 for Sam, so it must be 0.4 for Alex (the probabilities must add to 1):

Then fill out the branches for Sam (0.5 Yes and 0.5 No), and then for Alex (0.3 Yes and 0.7 No):

Now it is neatly laid out we could calculate probabilities (read more at “Tree Diagrams”).

## Mutually Exclusive

**Mutually Exclusive** means you can’t get both events at the same time.

It is either one or the other, but **not both**

Examples:

- Turning left or right are Mutually Exclusive (you can’t do both at the same time)
- Heads and Tails are Mutually Exclusive
- Kings and Aces are Mutually Exclusive

What isn’t Mutually Exclusive

- Kings and Hearts are
**not**Mutually Exclusive, because you can have a King of Hearts!

Like here:

Aces and Kings are Mutually Exclusive |
Hearts and Kings are Mutually Exclusivenot |

Read more at Mutually Exclusive Events