The sample space of an experiment is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} and A = {TTT}, B = {HTT, THT, TTH}, C = {HHT, HTH, THH, HHH},
then A, B and C form a set of?
a)
Mutually exclusive but not exhaustive events
b)
Exhaustive but not mutually exclusive events
c)
Mutually exclusive and exhaustive events
d)
Neither mutually exclusive nor exhaustive events
Correct answer is option 'C'. Can you explain this answer?

Question

The sample space of an experiment is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} and A = {TTT}, B = {HTT, THT, TTH}, C = {HHT, HTH, THH, HHH},
then A, B and C form a set of?

a)
Mutually exclusive but not exhaustive events
b)
Exhaustive but not mutually exclusive events
c)
Mutually exclusive and exhaustive events
d)
Neither mutually exclusive nor exhaustive events

Correct answer is option 'C'. Can you explain this answer?

Answer

Above information has no common elements
in three of them and U of three sets forms
sample set . Therefore option C is correct

Answer

C is correct of the science is say to be set is mutually exclusive and exhaustive events.

Answer

Explanation:

To determine whether events A, B, and C form a set of mutually exclusive and exhaustive events, we need to understand the definitions of these terms and analyze the given sample space.

Mutually Exclusive Events:
Two events are said to be mutually exclusive if they cannot occur at the same time. In other words, the occurrence of one event excludes the occurrence of the other event.

Exhaustive Events:
A set of events is said to be exhaustive if one of the events must occur. In other words, the events in the set cover all possible outcomes of the given experiment, leaving no room for any other outcome.

Analyzing the Sample Space:
The given sample space consists of 8 outcomes: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Let's check whether events A, B, and C satisfy the conditions for being mutually exclusive and exhaustive.

Event A: A = {TTT}
- This event consists of a single outcome, TTT.
- Since there are no other outcomes in the sample space that include three consecutive tails, event A is mutually exclusive with all other events.
- However, event A alone does not cover all possible outcomes of the experiment, as it only includes one outcome. Therefore, event A is not exhaustive.

Event B: B = {HTT, THT, TTH}
- This event consists of three outcomes: HTT, THT, and TTH.
- None of these outcomes overlap with any of the outcomes in event C.
- However, event B alone does not cover all possible outcomes of the experiment, as it only includes three outcomes. Therefore, event B is not exhaustive.

Event C: C = {HHT, HTH, THH, HHH}
- This event consists of four outcomes: HHT, HTH, THH, and HHH.
- None of these outcomes overlap with any of the outcomes in event B.
- Event C covers all possible outcomes of the experiment, as it includes all four outcomes that have at least two heads. Therefore, event C is exhaustive.
- Additionally, since there are no outcomes that belong to both event C and event A or B, event C is mutually exclusive with events A and B.

Conclusion:
Based on the analysis of the given sample space and events A, B, and C, we can conclude that events A, B, and C form a set of mutually exclusive and exhaustive events. Therefore, the correct answer is option 'C' - Mutually exclusive and exhaustive events.

Answer

Explanation:
To determine whether A, B, and C form a set of mutually exclusive and exhaustive events, we need to understand the definitions of these terms:

1. Mutually exclusive events: Two or more events are said to be mutually exclusive if they cannot occur simultaneously. In other words, if one event happens, the other event(s) cannot happen at the same time.

2. Exhaustive events: A set of events is said to be exhaustive if at least one of the events must occur. In other words, the set of events covers all possible outcomes.

Now, let's analyze the given events A, B, and C:

A = {TTT}
Event A represents getting three tails in a row. Since there are no other outcomes in the sample space S that contain three tails in a row, A is mutually exclusive from the other events.

B = {HTT, THT, TTH}
Event B represents getting exactly two tails in three coin flips. There are no overlapping outcomes between event B and event C, so B is mutually exclusive from C.

C = {HHT, HTH, THH, HHH}
Event C represents getting at least one head in three coin flips. There are no overlapping outcomes between event C and events A or B, so C is mutually exclusive from both A and B.

Mutually exclusive:
From the analysis above, we can see that events A, B, and C are mutually exclusive since no outcomes are shared between them.

Exhaustive:
To determine if events A, B, and C are exhaustive, we need to consider if all possible outcomes are covered by these events. The sample space S contains 8 possible outcomes, and when we combine the outcomes of events A, B, and C, we can account for all 8 outcomes:

A = {TTT} (1 outcome)
B = {HTT, THT, TTH} (3 outcomes)
C = {HHT, HTH, THH, HHH} (4 outcomes)

The total number of outcomes in A, B, and C is 1 + 3 + 4 = 8, which covers all the outcomes in the sample space S. Therefore, events A, B, and C are exhaustive.

Conclusion:
Based on the analysis above, we can conclude that events A, B, and C are both mutually exclusive and exhaustive. Therefore, the correct answer is option 'C'.

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