Lesson 1, Topic 1
In Progress

1.15. Use random events to explore and apply probability concepts in simple life

ryanrori February 3, 2021

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After completing this section, the learner will be able to use random events to explore and apply probability concepts in simple life, by successfully completing the following:

  • Gather, organise, sort and classify data in a suitable manner for further processing and analysis
  • Choose experiments and simulations appropriately in terms of the situation to be investigated
  • Determine probabilities correctly
  • Make distinctions correctly between theoretical and experimental probabilities
  • Base predictions on validated experimental or theoretical probabilities
  • Communicate the outcomes of experiments and simulations clearly

As human beings we are constantly trying to determine our future- we seek a means of measuring our potential for certain risks arising.  

The mathematics of probability is concerned with developing mathematical ways of determining the likelihood of an event occurring – in those cases where it is possible to determine this likelihood.  By contrast, most of our everyday (colloquial) predictions of likelihood are based on our instinct or gut-feel of how likely it is that an event will occur. 

Probability represents a measurement device.  Probabilities measure the likelihood of events occurring.  They might be applied to measuring the likelihood of death at a certain age, motor accidents, plane crashes, insolvencies, promotions, new employment, or even a new relationship.  

Random events, equal likelihood, probability

Here are some of the important concepts you must know:

TrialAn action that has a specific outcome. A “trial” is any action performed repeatedly that or may not lead to a different result every time. An example is a turn at rolling a die. Another example is the toss of a coin
OutcomeThe result obtained when performing a trial. One example is getting a 6 when you roll a die. Another example is getting “heads” when tossing a coin
EventAn event is a single occurrence. A trial and its outcome together make up an event. For example, getting a 3 when you have rolled a die is an event
LikelihoodLikelihood is measured on a scale between 0 and 1. Where it is absolutely certain an event will occur (the sun will rise tomorrow, new technologies will be developed, or the certainty of death are examples) the probability figure is 1.  An event with a probability ratio of 0,2 is less likely to occur than an event with a probability ratio of 0,95.
Equal likelihoodSome outcomes are equally likely to result from an action that is repeatedly performed. For example, rolling a die that has not been “doctored” (we use the term “fair”, or “unbiased”, like a referee that does not favour one team over the other) will definitely give an equal chance of producing a 1, or a 2, or a 3, or a 4, or a 5, or a 6Some outcomes are equally likely to result from an action that is repeatedly performed. For example, rolling a die that has not been “doctored” (we use the term “fair”, or “unbiased”, like a referee that does not favour one team over the other) will definitely give an equal chance of producing a 1, or a 2, or a 3, or a 4, or a 5, or a 6
Probability- the formula to calculate probabilityProbability is about estimating how likely (probable) something is to happen. Probability can be used to predict, for example, the outcome when throwing a die or tossing a coin, The probability that some event will take place is expressed in mathematics as a number between 0 and 1. A probability of 0 (zero) means that it is impossible for a particular event to occur. The probability that the sun will not rise tomorrow is 0. On the other hand, the probability that a ball thrown upwards will return to the ground is 1. A probability of 1 indicates that an event is sure—absolutely certain—to occur. If a fair coin is tossed, then the probability of the outcome being “heads” is 0, 5.We calculate probability from the formula:

Gather, organise, sort and classify data for processing and analysis

In 1.1 and 1.2, we looked at the five steps of conducting research, which entailed gathering, organising, sorting and classifying data in a suitable manner for further processing and analysis. 

We will now continue by looking at experiments and simulations as methods of gathering, organising, sorting and classifying data.

Choose experiments and simulations appropriately

All statistical experiments have three things in common:

The experiment can have more than one possible outcome. Each possible outcome can be specified in advance. The outcome of the experiment depends on chance. 

A coin toss has all the attributes of a statistical experiment. There is more than one possible outcome. We can specify each possible outcome (i.e., heads or tails) in advance. And there is an element of chance, since the outcome is uncertain. 

  • A sample space is a set of elements that represents all possible outcomes of a statistical experiment. 
  • A sample point is an element of a sample space. 
  • An event is a subset of a sample space – one or more sample points: 
  • Two events are mutually exclusive if they have no sample points in common. 
  • Two events are independent when the occurrence of one does not affect the probability of the occurrence of the other. 

Rolling a Die: Examples

  1. Suppose I roll a die. Is that a statistical experiment? 

Yes. Like a coin toss, rolling dice is a statistical experiment. There is more than one possible outcome. We can specify each possible outcome in advance. And there is an element of chance. 

  1. When you roll a single die, what is the sample space? 

The sample space is all of the possible outcomes – an integer between 1 and 6. 

  1. Which of the following are sample points when you roll a die – 3, 6, and 9? 

The numbers 3 and 6 are sample points, because they are in the sample space. The number 9 is not a sample point, since it is outside the sample space; with one die, the largest number that you can roll is 6. 

  1. Which of the following sets represent an event when you roll a die? 

A.   {1}

B.   {2, 4,}

C.   {2, 4, 6}

D.   All of the above 

The correct answer is D. Remember that an event is a subset of a sample space. The sample space is any integer from 1 to 6. Each of the sets shown above is a subset of the sample space, so each represents an event. 

  1. Consider the events listed below. Which are mutually exclusive? 

A.   {1}

B.   {2, 4,}

C.   {2, 4, 6} 

Two events are mutually exclusive, if they have no sample points in common. Events A and B are mutually exclusive, and Events A and C are mutually exclusive; since they have no points in common. Events B and C have common sample points, so they are not mutually exclusive. 

  1. Suppose you roll a die two times. Is each roll of the die an independent event? 

Yes. Two events are independent when the occurrence of one has no effect on the probability of the occurrence of the other. Neither roll of the die affects the outcome of the other roll; so each roll of the die is independent.