Probability

By:Ericson David A. Guerrero

BSED 2 Irr Mathematics Major

Partido State University

College Of Education

Ericsondavid94@gmail.com

'Probability is the bane of the age,' said Moreland, now warming up. 'Every Tom, Dick, and Harry thinks he knows what is probable. The fact is most people have not the smallest idea what is going on round them. Their conclusions about life are based on utterly irrelevant - and usually inaccurate - premises.'

How likely something could happen.

Many events can't be predicted with total certainty. The best we can do is say how likely they are to happen, using the idea of probability.

Tossing a Coin  When a coin is tossed, there are two possible outcomes: heads (H) or tails (T) We say that the probability of the coin landing H is 1/2. And the probability of the coin landing T is 1/2.

Throwing Dice  When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6.

Probability 

In general:

Probability of an event happening =		Number of ways it can happen
		Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability =	1
	6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble will be picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability =	4	= 0.8
	5

Probability Line

You can show probability on a Probability Line:

The 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 1/2 chance, so we would expect 50 Heads.

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

Words

Some words have special meaning in Probability:

Experiment: an action where the result is uncertain.

Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.

Sample Space: all the possible outcomes of an experiment

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

·         the 5 of Clubs is a sample point

·         the King of Hearts is a sample point

"King" is not a sample point. As there are 4 Kings that is 4 different sample points.

Event: a single result of an experiment

Example Events:

·         Getting a Tail when tossing a coin is an event

·         Rolling a "5" is an event.

An event can include one or more possible outcomes:

·         Choosing a "King" from a deck of cards (any of the 4 Kings) is an event

·         Rolling an "even number" (2, 4 or 6) is also an event

The Sample Space is all possible outcomes. A Sample Point is just one possible outcome. And an Event can be one or more of the possible outcomes.

Hey, let's use those words, so you get used to them:

Example: Alex decide to see how many times a "double" would come up when throwing 2 dice.

Each time Alex throws the 2 dice is an Experiment.

It is an Experiment because the result is uncertain.

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

The Sample Space is all possible outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

These are Alex's Results:

Experiment	Is it a Double?
{3,4}	No
{5,1}	No
{2,2}	Yes
{6,3}	No
...	...

After 100 Experiments, Alex had 19 "double" Events ... is that close to what you would expect?

Reference:

http://www.mathsisfun.com/data/probability.html