Probability, Outcomes and Sample Space

Probability, Outcomes and Sample Space

Probability describes the chance that an uncertain event will occur. Another way of saying this is that probability tells how often a certain event will happen (or not happen)

Probability is a shown by a number between 0 and 1. a probability of 0 (zero) means an event will never happen. A probability of 1 (one) means an event will always happen.

Probability = ^{number of ways an event can occur}

_{total number of outcomes}

There are two ways to talk about or determine probability, Theoretical Probability and Experimental Probability.

Theoretical Probability

Theoretical probability is when all of the number of ways an event can occur is written over (divided by) all of the outcomes (events) which can occur.

Sample event: Rolling a on a 6 sided die

Outcomes: All of the numbers that can be rolled on a die, for example,

This shows one (1) event, the number 3 on a die and six (5) possible outcomes, 1, 2, 3, 4, 5, and 6.

Therefore, the probability of rolling a 3 on a die is 1 out of 6.

P(3)= ¹

₆

Experimental probability

Experimental probability is when the number of times an event occurs is written over (divided by) the number of trials conducted.

Determine the probability of rolling a by comducting an experiment. Roll the dice 24 times:

Count the number of times the event (rolling a ) occurs. Write that result over the number of times the experiment was done (24 rolls). This is the experimental probability.

P(exp)= ⁴

₂₄

In the examples above, the theoretical probability of rolling a specific number on a single 6-sided die is 1/6. This can be changed to a decimal. 1 ÷ 6 = 0.1666 (a repeating decimal) or multiply it by 100 and make it 16.6(repeating)%.

In the example of experimental probability, the result is 4/24 which reduces to 1/6. However, in the real world, things are not so perfect.

EXPERIMENT: Roll a number cube and keep track of the number it lands on for each roll. If everything was perfect, each number would land the same number of times as the others. The study of probability shows that the results from expeiments will come close to the theoretical probability only if a large number of trials are done.

After trying this experiment with a die, try this program. enter how many dice rolls you want to do. (The bigger the number, the longer it takes). When the new window opens, scroll down and see what the experimental probability was for each number.

Simulated Experimental Dice-Roll Data

How many dice rolls?

The results of the dice rolls will appear in a pop-up window. If you have pop-ups disabled, you might have to check to see if another window opened in the background.

©Jeff LeMieux, December 2004

To review, an event is a particular outcome or set of outcomes that is chosen. In one of the examples above a was chosen as the event. Outcomes are all the possible ways all the possible events can occur, for this example, all the possible events are .

This list of all possible outcomes is called a sample space. all the items in a sample space must be mutually exclusive - which means only ONE event can occur at a time. When rolling a die, you can only get one number at a time, therefore the outcomes for a die are mutually exclusive.

Here are some examples of sample space and mututally exclusive events:

Event: Roll a 6-side number cube
Mutually exclusive sample space:
Event: Coin toss
Mutually exclusive sample space:
Event: 4-sector spinner
Mutually exclusive sample space:

biased and unbiased = fair and unfair

©J. LeMieux, December 2004

Probability =	^{number of ways an event can occur}
	_{total number of outcomes}