Sucesos elementales.

	Probability and chance: elementary events.
	3rd year of secondary education. (Probability).

Equally likely outcomes.

In this window you can see a simulation of a tetrahedral die. It has four sides and the number which appears on its base is the number which is thrown. Imagine that we have thrown the die once and get the number indicated in the window below. Follow the instructions in the margin and answer the questions that follow.

Look carefully at the number which appears on the base of the tetrahedron. This is the number that we get when we "throw" the die.

Next, click on the blue arrow next to the number which has been thrown on the die.

Each time you click on the relevant blue arrow the result is added to the table and the die is "thrown" again.

1. Throw the four-sided die in the above window 50 times and look carefully at the figures indicating the absolute and relative frequency (don't delete the results, i.e. don't click on the init button).
- Which number has the highest relative frequency? Which has the lowest?
- Work out the difference between the highest and lowest relative frequency.

2. Throw the die another 50 times until you've thrown it 100 times in total. Look at the absolute and relative frequency figures again.
- What is the probability of throwing a 1 with this die? What about throwing a 2? what about a 3? and a 4?
- Which number has the highest probability of being thrown? (don't delete the information)

3. Write the results given in the table in your exercise book and work out the percentages indicating the number of times each number was thrown.
- Is there a lot of difference between the percentages?

4. Imagine that the same exercise is carried out by all the classes in your school and that afterwards all the results are put together. What do you think the results would be like? What do you think they depend on?

We have been playing with a virtual die. If we had done the same exercise with a real die we would have got similar results. This leads us to the following conclusion:

Any number on a die is just as likely to be thrown as another. Therefore, we can say that each number has the same probability of being thrown or that they are equally likely outcomes.

Outcomes which are not equally likely.

In this section we are going to start by focusing on a motor race. The following windows illustrate two dice being thrown and the cars in the race.

In this game the cars move in the following way: throw the two dice and move the car, whose number corresponds to the total of the two dice, forward one square by dragging it with the mouse. PLAY THE GAME AND SEE WHO WINS!

Once you've finished the game write down which car has won and look carefully at the position of each car at the end of the race. Do you think that each car has the same probability of winning?

	Look carefully at this table and try to relate the information in it to the results of the game.
When two dice are thrown each possible total score does NOT have the same probability of occurring. Therefore, we can say that the events are NOT EQUALLY LIKELY to occur.

Laplace's rule.

A box contains 10 equal-sized balls of different colours.
An experiment is carried out in which a ball is drawn out of the box at random (without looking)

Think about the following events. Selecting:
a) a red ball
b) a green ball
c) a red, yellow or brown ball
d) a blue or green ball
e) a yellow ball

Draw the balls out of the box in this window by dragging them with the mouse.

Which of the events above are equally likely to occur?

Imagine that you draw a ball out of the box at random a million times.

- How many times, approximately, do you think that each colour would be selected?
- What fraction of the total number would this number represent?

If we focus on the event "selecting a red ball" we can refer to the number of red balls in the box as the "number of successful outcomes" (giving the required result) and the total number of balls in the box are referred to as the "number of possible outcomes".

The THEORETICAL PROBABILITY of event A, written as p(A), happening is:
This method of calculating the probability of an event happening is known as Laplace's rule.	In order to apply this rule to an event all the possible outcomes must be equally likely to occur.

Therefore the probability of selecting a red ball from the box above is:

and the probability of selecting a green ball is:

Likewise p(yellow ball) = p(blue ball) = p(brown ball) = 0.2

-What is the probability of selecting a blue or green ball? (First look at the number of successful outcomes there are now by taking the balls that we are interested in out of the box)

-What is the probability of selecting a red, yellow or brown ball? (Look at the number of successful outcomes there are now by taking the balls that we are interested in out of the box)

Exercise 1

What is the probability of the following events happening when the tetrahedral, or four-sided, die is thrown? (refer back to the window at the top of this page):

a) Throwing a 3
b) Throwing an even number
c) Throwing a number greater than 1
d) Throwing an 8
e) Throwing a number lower than 5

Certainty and impossibility.

You should have noticed that the answer to question d) in the exercise above was zero.

In other words, the probability of throwing an 8 with a four-sided die is zero, as there are no successful outcomes. Therefore, we can say that it is an impossible event and its probability is zero.

However, the answer to question e) is one as all the outcomes are successful because when you throw a four-sided die all the numbers thrown are lower than 5. Therefore, we can say that it is a certain event and its probability is one.

You will have also noticed that all the other probabilities that you calculated fell between zero and one.

Exercise 2

When two six-sided dice are thrown what is the probability that the total number thrown is the following?: (refer back to the dice table above)

an even number

a multiple of 3

	Ángela Núñez Castaín

	Spanish Ministry of Education. Year 2001

Except where otherwise noted, this work is licensed under a Creative Common License