Question

Discuss the following concepts and give examples from everyday life in which you might encounter each concept. Hint: For instance, consider the “experiment” of arriving for class. Some possible outcomes are not arriving (that is, missing class), arriving on time, and arriving late.

• (a)

  Sample space.

• (b)

  Probability assignment to a sample space. In your discussion, be sure to include answers to the following questions.

  • (i)

    Is there more than one valid way to assign probabilities to a sample space? Explain and give an example.

  • (ii)

    How can probabilities be estimated by relative frequencies? How can probabilities be computed if events are equally likely?

Answer 1

Solution:

a. ​​​​​​Sample Space: It is a set of all possible outcomes from a given experiment or an event. It is usually denoted by 'S'.

Every day example include:

1.If we want to play music of three different genres i.e. rock , classical and pop in the mobile phone and we select the shuffle

button, the outcome of the next music played will be either pop or rock or classical.

2.If one wants to eat a fruit and at home there are apples, oranges or banana. Then the outcome will be the person will either eat an apple or an orange or a banana.

b. Probability assignment to a sample space: Once the sample space ,that is all possible outcomes is listed we move on to assigning probability to each particular outcomes.These probabilities show the chances or the likelihood of the particular outcome's occurrence.

In the example 1 above if there are a total of 20 songs in the playlist having 5 classical musics , 10 rock musics and 5 pop musics. Then the probability that the first music played will be either classical having a probability of 5/20 or rock music    having a probability of 10/20 or pop music having a probability of 5/20.The probabilities are calculated by dividing the total number of particular music genre by total number of musics in the playlist.

i.There are more than one way of assigning probabilities.The three different ways of assigning probabilities are :

- Classical approach: Here the chances of occurrence of the different outcome is equal.

Example: Probability of getting up early , on time or late will be equally likely i.e.1/3 ,1/3 and 1/3 for the respective outcomes.

-Relative frequency approach:Here the past or historic data or the results of an experiment is taken into consideration to decide the probabilities.

Example:Probability of passing in English with grade A can be decided from a student's past performance in the same subject

- Subjective approach:Here the degree of belief or judgement is used as the basis for assigning the probabilities.

Example: Probability of passing in English with grade A can also be decided based on the judgement from the amount of hardwork done, the level of preparation , the difficulty level of the question paper etc.

ii.As explained above probabilities can be assigned using past /historic data,or from the results of an experiment .

Example:Referring to the Relative frequency approach example above, if the past 10 exam results shows that the student has got grade A in 8 out of the past 10 exams in English, then the probability that the student will get grade A is 8/10 = 0.8 i.e 80% chance of getting grade A.

  To compute probability for equally likely event the formula =1/ total no. of outcomes of a particular event.

Example : Referring to the Classical approach example above the outcomes of getting up is early, on time or late. So total outcomes are three. Thus the probability of each outcome occurring is 1/3 , 1/3 and 1/3 respectively.

Hope I was able to answer your question.

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