Mathematics : Topic 5: Probability

What is Probability?

Probabilities are written as fraction or decimals, and less often as percentage.

An event can have several possible outcomes.

Each outcome has a probability or chance of occurring.

When a fair dice is thrown there is equal chance of throwing each number. The outcomes from the event throwing a dice are equally likely outcomes.

If the outcomes of an event are equally likely, the probability can be calculated using.

Formula Probability

Probability of an event = Number of successful outcomes

Total number of possible outcomes

The Probability Scale

· Probability is measured on a scale from 0 to 1

· If an event is impossible, it has a probability of 0

· If an event is certain, it has a probability of 1

Example:

1. A bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. What is the probability of pulling a green marble from te bag without looking?

There are 3 green marbles out of 10

P(green) = 3/10

2. A fair dice rolled.

What is the probability of rolling:

a. 5? b. an even number 1

a) There are 6 numbers on a dice. One of them is a 5.

P (5) = 1/ 6

b) There are 3 even number.

P (even) = 3/6 = ½

Probability Line

We can show probability on a Probability Line:

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

· Example: toss a coin 100 times, how many Heads will come up?

· Probability says that heads have a ½ chance, so we can expect 50 Heads.

· But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Probability: Types of Events

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

Events can be:

Independent (each event is not affected by other events),
Dependent (also called "Conditional", where an event is affected by other events)
Mutually Exclusive (events can't happen at the same time)

Independent Events

· Events can be:

Independent (each event is not affected by other events),
Dependent (also called "Conditional", where an event is affected by other events)
Mutually Exclusive (events can't happen at the same time)

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.

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 look at 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 we are removing cards from the deck.

Replacement: When we 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.

Tree Diagrams

When we 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 can calculate probabilities

Mutually Exclusive

Mutually Exclusive means we 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