Discuss The Concept Of Probability
Discuss The Concept Of Probability, Including The Notion Of An Experiment, Outcomes, And Sample Space. How Can Each Of These Be Useful To Us?
Probability is a number that determines the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%. In this essay, I shall examine experiments whose results I cannot predict in advance. Such experiments are called Random Experiments. If I toss a coin, I may not be able to say for sure whether the coin will land with the head up or with the tail up. Similarly, if I toss a fair die, I am sure it will land but I am not sure which of the six faces will show up. The experiments of tossing a fair coin or a fair die are examples of random experiments. Probability is a branch of Mathematics dealing with random experiments.
Suppose I toss a fair coin 100 times, experience has shown that in most instances, the number of times a head shows up will be very close to 50. If I toss the coin 1000 times, the number of times a head shows up will also be very close to 500. If I denote the number of times we are successful in getting a head showing up by n(s) i.e. number of successes, and we denote the total number of trials by n(t), from empirical results, the ratio becomes steady as the number of trials becomes large indefinitely. The ratio defines the probability that a head shows up. The result is strictly empirical but it is very important, as it forms the basis of the theoretical work in probability.
SAMPLE SPACE AND EVENT SPACE
Any result of an experiment in probability is usually called an OUTCOME. If we cannot predict beforehand, the outcome of an experiment the experiment is called a RANDOM EXPERIMENT. The set of all possible outcomes of any random experiment will be called a SAMPLE SPACE and it will be denoted by S. The number of outcomes in S or the number of elements in the sample space will be denoted n(s).
A subset of the sample space which may be a collection of outcomes of a random experiment is called an EVENT SPACE. I will denote an event space by E and the number of outcomes or elements in E by n(E).
The probability of an event E denoted Pr(E) is defined as Pr(E) =
Since the empty set is a subset of the sample space n() = 0. Pr() = = 0
Pr() is the probability of an impossible event. Since s is a subset of itself. Pr() = = 1
Pr(s) is the probability of an event which is certain to occur. The probability of an event E is, therefore, a number which satisfies the inequality 0 ≤ Pr(E) ≤1.
EXAMPLE
I was required to complete this specific task where I was to obtain a desired outcome of the task. A random sample of sixty candidates who sat for CCNA I and II of the CISCO examinations in a certain year is taken with the table that shows the number of candidates who passed or failed each part of the examination.
|
CCNA I |
Total | |||
|
Pass |
Fail | |||
|
CCNA II |
Pass Fail | |||
|
20 |
35 | |||
|
Total |
24 |
60 | ||
Completing the table, I answered these following questions and evaluated the ideal experiment outcome and sample space.
- If a candidate is chosen at random from the sample, use the table to find the probability that the candidate:
- Passed CCNA I
- Passed CCNA I and II
- Passed CCNA II but failed CCNA I.
- If a candidate is chosen at random from the subgroup of those who failed CCNA I, find the probability that the candidate passed CCNA II.
SOLUTION
|
CCNA I |
Total | |||
|
Pass |
Fail | |||
|
CCNA II |
Pass Fail |
9 |
16 |
25 |
|
Total |
24 |
36 |
60 | |
i. let be the event that a candidate chosen at random passed CCNA II.
n() = 25; n(S)= 60
Pr() = = =
- Let be the event that a candidate chosen at random passed CCNA I and II
n() = 9; n(S)= 60
Pr() = = =
iii. Let be the event that a candidate chosen at random passed CCNA II but failed CCNA I
n() = 16; n(S)= 60
Pr() = = =
- If a candidate is chosen from a subgroup of those who failed CCNA I then the sample space has changed.
Let S’ be the new sample space: n(S’) = 36.
Let be the event that a candidate is chosen at random from a subgroup of those that failed CCNA I but passed CCNA II, n() = 16
n() = 16; n(S’)= 36
Pr() = = =
Through the use of probability, I was able to solve the required task as a fraction and also as a percentage.
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