Which Pair Does Not Represent The Probabilities Of Complementary Events?A. 4 7 \frac{4}{7} 7 4 ​ And 3 7 \frac{3}{7} 7 3 ​ B. 16 % 16\% 16% And 84 % 84\% 84% C. 0.25 0.25 0.25 And 0.50 0.50 0.50 D. 0 0 0 And 1 1 1

Introduction

In probability theory, complementary events are pairs of events that are mutually exclusive and exhaustive. This means that one event cannot occur while the other event occurs, and together they cover all possible outcomes. Complementary events are often denoted as A and A', where A is the original event and A' is its complement. The probability of a complementary event is calculated as the difference between 1 and the probability of the original event. In this article, we will explore which pair of probabilities does not represent the probabilities of complementary events.

What are Complementary Events?

Complementary events are pairs of events that are mutually exclusive and exhaustive. This means that one event cannot occur while the other event occurs, and together they cover all possible outcomes. For example, consider a coin toss. The event of getting heads (H) is a complementary event to the event of getting tails (T). If we denote the event of getting heads as A, then the event of getting tails is A'. The probability of getting heads is 0.5, and the probability of getting tails is also 0.5.

Calculating Probabilities of Complementary Events

The probability of a complementary event is calculated as the difference between 1 and the probability of the original event. Mathematically, this can be represented as:

P(A') = 1 - P(A)

where P(A') is the probability of the complementary event and P(A) is the probability of the original event.

Analyzing the Options

Now, let's analyze the options given in the problem:

A. $\frac{4}{7}$ and $\frac{3}{7}$

B. $16\%$ and $84\%$

C. $0.25$ and $0.50$

D. $0$ and $1$

We need to determine which pair of probabilities does not represent the probabilities of complementary events.

Option A: $\frac{4}{7}$ and $\frac{3}{7}$

Let's calculate the probability of the complementary event for the first option:

P(A') = 1 - P(A) = 1 - $\frac{4}{7}$ = $\frac{3}{7}$

This means that the probability of the complementary event is $\frac{3}{7}$, which is the same as the probability of the original event. Therefore, this pair of probabilities does represent the probabilities of complementary events.

Option B: $16\%$ and $84\%$

Let's calculate the probability of the complementary event for the second option:

P(A') = 1 - P(A) = 1 - $16\%$ = $84\%$

This means that the probability of the complementary event is $84\%$, which is the same as the probability of the original event. Therefore, this pair of probabilities does represent the probabilities of complementary events.

Option C: $0.25$ and $0.50$

Let's calculate the probability of the complementary event for the third option:

P(A') = 1 - P(A) = 1 - $0.25$ = $0.75$

This means that the probability of the complementary event is $0.75$, which is not the same as the probability of the original event. Therefore, this pair of probabilities does not represent the probabilities of complementary events.

Option D: $0$ and $1$

Let's calculate the probability of the complementary event for the fourth option:

P(A') = 1 - P(A) = 1 - $0$ = $1$

This means that the probability of the complementary event is $1$, which is the same as the probability of the original event. Therefore, this pair of probabilities does represent the probabilities of complementary events.

Conclusion

In conclusion, the pair of probabilities that does not represent the probabilities of complementary events is:

C. $0.25$ and $0.50$

This is because the probability of the complementary event is $0.75$, which is not the same as the probability of the original event.

Final Answer

Introduction

In our previous article, we explored the concept of complementary events in probability theory. Complementary events are pairs of events that are mutually exclusive and exhaustive, and their probabilities are calculated as the difference between 1 and the probability of the original event. In this article, we will answer some frequently asked questions about complementary events in probability.

Q: What are complementary events?

A: Complementary events are pairs of events that are mutually exclusive and exhaustive. This means that one event cannot occur while the other event occurs, and together they cover all possible outcomes.

Q: How are the probabilities of complementary events calculated?

A: The probability of a complementary event is calculated as the difference between 1 and the probability of the original event. Mathematically, this can be represented as:

P(A') = 1 - P(A)

where P(A') is the probability of the complementary event and P(A) is the probability of the original event.

Q: What is the relationship between complementary events and mutually exclusive events?

A: Complementary events are a special case of mutually exclusive events. Mutually exclusive events are events that cannot occur at the same time, and complementary events are pairs of mutually exclusive events that cover all possible outcomes.

Q: Can complementary events be independent?

A: No, complementary events cannot be independent. By definition, complementary events are pairs of events that are mutually exclusive and exhaustive, which means that they are not independent.

Q: Can complementary events be dependent?

A: Yes, complementary events can be dependent. For example, consider a coin toss. The event of getting heads (H) is a complementary event to the event of getting tails (T). If we denote the event of getting heads as A, then the event of getting tails is A'. The probability of getting heads is 0.5, and the probability of getting tails is also 0.5. In this case, the events are dependent because the outcome of one event affects the probability of the other event.

Q: Can complementary events be equally likely?

A: Yes, complementary events can be equally likely. For example, consider a coin toss. The event of getting heads (H) is a complementary event to the event of getting tails (T). If we denote the event of getting heads as A, then the event of getting tails is A'. The probability of getting heads is 0.5, and the probability of getting tails is also 0.5. In this case, the events are equally likely.

Q: Can complementary events be unlikely?

A: Yes, complementary events can be unlikely. For example, consider a deck of cards. The event of drawing a red card is a complementary event to the event of drawing a black card. If we denote the event of drawing a red card as A, then the event of drawing a black card is A'. The probability of drawing a red card is 0.5, and the probability of drawing a black card is also 0.5. In this case, the events are equally likely.

Q: Can complementary events be certain?

A: No, complementary events cannot be certain. By definition, complementary events are pairs of events that are mutually exclusive and exhaustive, which means that they cannot be certain.

Q: Can complementary events be impossible?

A: No, complementary events cannot be impossible. By definition, complementary events are pairs of events that are mutually exclusive and exhaustive, which means that they cannot be impossible.

Conclusion

In conclusion, complementary events in probability are pairs of events that are mutually exclusive and exhaustive, and their probabilities are calculated as the difference between 1 and the probability of the original event. We hope that this Q&A guide has helped to clarify any questions you may have had about complementary events in probability.

Final Answer

The final answer is that complementary events are pairs of events that are mutually exclusive and exhaustive, and their probabilities are calculated as the difference between 1 and the probability of the original event.