Probability

Objective:

This topic gives an overview of; 

  • Chance and Probability
  • Getting a Result
  • Equally likely outcomes
  • Linking chances to probability
  • Outcomes as events
  • Chance and probability related to real life

Chance and Probability

Sometimes it happens that during rainy season, you carry a raincoat every day and it does not rain for many days. However, by chance, one day you forget to take the raincoat and it rains heavily on that day.

Sometimes it so happens that a student prepares 4 chapters out of 5, very well for a test. But a major question is asked from the chapter that she left unprepared.

Everyone knows that a particular train runs in time but the day you reach well in time it is late! You face a lot of situations such as these where you take a chance and it does not go the way you want it to. These are examples where the chances of a certain thing happening or not happening are not equal. The chances of the train being in time or being late are not the same. When you buy a ticket which is wait listed, you do take a chance. You hope that it might get confirmed by the time you travel.

We however, consider here certain experiments whose results have an equal chance of occurring.

Getting a Result

You might have seen that before a cricket match starts, captains of the two teams go out to toss a coin to decide which team will bat first.

What are the possible results you get when a coin is tossed? Of course, Head or Tail. Imagine that you are the captain of one team and your friend is the captain of the other team. You toss a coin and ask your friend to make the call. Can you control the result of the toss? Can you get a head if you want one? Or a tail if you want that? No, that is not possible. Such an experiment is called a random experiment. Head or Tail are the two outcomes of this experiment.

Equally Likely Outcomes

A coin is tossed several times and the number of times we get head or tail is noted. Let us look at the result sheet where we keep on increasing the tosses:

Observe that as you increase the number of tosses more and more, the number of  heads and the number of tails come closer and closer to each other.This could also be done with a die, when tossed a large number of times. Number of each of the six outcomes become almost equal to each other. In such cases, we may say that the different outcomes of the experiment are equally likely. This means that each of the outcomes has the same chance of occurring.

Linking Chances to Probability

Consider the experiment of tossing a coin once. There are only two outcomes – Head or Tail. Both the outcomes are equally likely.

Likelihood of getting a head is one out of two outcomes, i.e., «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math». In other words, we say that the probability of getting a head =«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math».

Now take the example of throwing a die marked with 1, 2, 3, 4, 5, 6 on its faces (one number on one face). If you throw it once, the outcomes are: 1, 2, 3, 4, 5, 6. Thus, there are six equally likely outcomes.

What is the probability of getting the outcome ‘2’?

It is   «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mfrac»«mo»§#8592;«/mo»«mo»§#8592;«/mo»«/mfrac»«mfrac»«mrow»«mo»§nbsp;«/mo»«mtext»Number«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»outcomes«/mtext»«mo»§nbsp;«/mo»«mtext»giving«/mtext»«mo»§nbsp;«/mo»«mtext»2«/mtext»«/mrow»«mrow»«mtext»Number«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»equally«/mtext»«mo»§nbsp;«/mo»«mtext»likely«/mtext»«mo»§nbsp;«/mo»«mtext»outcomes«/mtext»«/mrow»«/mfrac»«/math»

Outcomes as Events

Each outcome of an experiment or a collection of outcomes make an event. For example in the experiment of tossing a coin, getting a Head is an event and getting a Tail is also an event.

In case of throwing a die, getting each of the outcomes 1, 2, 3, 4, 5 or 6 is an event Is getting an even number an event? Since an even number could be 2, 4 or 6, getting an even number is also an event.

It is  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»3«/mn»«mn»6«/mn»«/mfrac»«mfrac»«mo»§#8592;«/mo»«mo»§#8592;«/mo»«/mfrac»«mfrac»«mrow»«mtext»Number«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»outcomes«/mtext»«mo»§nbsp;«/mo»«mtext»that«/mtext»«mo»§nbsp;«/mo»«mtext»make«/mtext»«mo»§nbsp;«/mo»«mtext»the«/mtext»«mo»§nbsp;«/mo»«mtext»event«/mtext»«/mrow»«mrow»«mtext»Total«/mtext»«mo»§nbsp;«/mo»«mtext»number«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»outcomes«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»the«/mtext»«mo»§nbsp;«/mo»«mtext»experiment«/mtext»«/mrow»«/mfrac»«/math»

Chance and Probability related to Real life

We talked about the chance that it rains just on the day when we do not carry a rain coat. Could it be one in 10 days during a rainy season? The probability that it rains is then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»1«/mn»«mn»10«/mn»«/mfrac»«/math» . The probability that it does not rain =«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»9«/mn»«mn»10«/mn»«/mfrac»«/math» . (Assuming raining or not raining on a day are equally likely) The use of probability is made in various cases in real life.

  1. To find characteristics of a large group by using a small part of the group.For example, during elections ‘an exit poll’ is taken. This involves asking the people whom they have voted for, when they come out after voting at the centres which are chosen off hand and distributed over the whole area. This gives an idea of chance of winning of each candidate and predictions are made based on it accordingly.
  2. Metrological Department predicts weather by observing trends from the data over many years in the past.

Summary:

  • There are certain experiments whose outcomes have an equal chance of occurring.
  •  A random experiment is one whose outcome cannot be predicted exactly in advance.
  • Outcomes of an experiment are equally likely if each has the same chance of occurring.
  • «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtext»Probability«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»an«/mtext»«mo»§nbsp;«/mo»«mtext»event«/mtext»«mo»§nbsp;«/mo»«mtext»=«/mtext»«mfrac»«mrow»«mtext»Number«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»outcomes«/mtext»«mo»§nbsp;«/mo»«mtext»that«/mtext»«mo»§nbsp;«/mo»«mtext»make«/mtext»«mo»§nbsp;«/mo»«mtext»an«/mtext»«mo»§nbsp;«/mo»«mtext»event«/mtext»«/mrow»«mrow»«mtext»Total«/mtext»«mo»§nbsp;«/mo»«mtext»number«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»outcomes«/mtext»«mo»§nbsp;«/mo»«mtext»of«/mtext»«mo»§nbsp;«/mo»«mtext»the«/mtext»«mo»§nbsp;«/mo»«mtext»experiment«/mtext»«/mrow»«/mfrac»«/math» when the outcomes are equally likely.
  • One or more outcomes of an experiment make an event.
  • Chances and probability are related to real life.

 

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