What is the sum of the probabilities of all the distinct outcomes within the same sample space ?a)3b)0c)1d)2Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Answer

Lets prove the answer with an example.
suppose there is a bag containing 9 ball in which 3 are red, 4 are blue and 2 balls are green.
Here we go,
Sample space n(S)=9
Event of getting red ball n(R)=3
Event of getting blue ball n(B)=4
Event of getting green ball n(G)=2
now
P(R)=n(R)/n(S)=3/9
P(B)=n(B)/n(S)=4/9
P(G)=n(G)/n(S)=2/9

P(R)+P(B)+P(G)=3/9 +4/9 +2/9
=(3+4+2)/9
=9/9
= 1
Here is your answer.

Answer

Understanding Probability and Sample Space
In probability theory, a sample space is the set of all possible outcomes of a particular experiment. The sum of the probabilities of all distinct outcomes in a sample space is a fundamental concept.
Sum of Probabilities
- The total probability of all outcomes in a sample space must always equal 1.
- This is a key axiom in probability theory, reflecting that one of the outcomes must occur when an experiment is performed.
Why is the Sum Equal to 1?
- Each distinct outcome in the sample space represents a particular scenario.
- When considering all possible outcomes, the cumulative probability accounts for every potential occurrence.
- Since the event must result in one of these outcomes, the total must reach unity (1).
Implications of the Sum Being 1
- If the sum were less than 1, it would imply that there are possible outcomes that have not been accounted for.
- Conversely, if the sum exceeds 1, it would suggest an impossibility, indicating an error in defining probabilities.
Conclusion
Thus, the correct answer to the question about the sum of the probabilities of all distinct outcomes within the same sample space is indeed option 'C': 1. This reflects the complete certainty that one of the outcomes will occur when the experiment is conducted.

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