If $P(A) = 0.24$ And $P(B) = 0.52$, And A And B Are Independent, What Is $P(A \text{ Or } B$\]?(a) 0.1248 (b) 0.28 (c) 0.6352 (d) 0.76 (e) The Answer Cannot Be Determined From The Information Given.

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Introduction

Probability is a fundamental concept in mathematics that deals with the study of chance events. It is used to measure the likelihood of an event occurring. In this article, we will explore the concept of probability and independence, and how to calculate the probability of two events occurring together.

What is Probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. The probability of an event A is denoted by P(A) and is calculated as the number of favorable outcomes divided by the total number of possible outcomes.

What is Independence?

Two events A and B are said to be independent if the occurrence of one event does not affect the probability of the other event. In other words, the probability of event A occurring is not affected by the occurrence or non-occurrence of event B. Mathematically, this can be represented as:

P(A ∩ B) = P(A) × P(B)

where P(A ∩ B) is the probability of both events A and B occurring together.

Given Information

We are given that P(A) = 0.24 and P(B) = 0.52, and that events A and B are independent. We are asked to find the probability of event A or event B occurring.

Calculating the Probability

To calculate the probability of event A or event B occurring, we can use the formula:

P(A or B) = P(A) + P(B) - P(A ∩ B)

Since events A and B are independent, we can substitute P(A ∩ B) = P(A) × P(B) into the formula:

P(A or B) = P(A) + P(B) - P(A) × P(B)

Now, we can plug in the given values:

P(A or B) = 0.24 + 0.52 - (0.24 × 0.52)

Simplifying the Expression

To simplify the expression, we can first calculate the product of 0.24 and 0.52:

0.24 × 0.52 = 0.1248

Now, we can substitute this value back into the expression:

P(A or B) = 0.24 + 0.52 - 0.1248

Evaluating the Expression

To evaluate the expression, we can first add 0.24 and 0.52:

0.24 + 0.52 = 0.76

Now, we can subtract 0.1248 from this result:

0.76 - 0.1248 = 0.6352

Conclusion

Therefore, the probability of event A or event B occurring is 0.6352.

Answer

The correct answer is (c) 0.6352.

Discussion

This problem illustrates the concept of independence in probability. When two events are independent, the probability of both events occurring together is equal to the product of their individual probabilities. This can be used to calculate the probability of two events occurring together, as shown in this example.

Additional Examples

Here are a few additional examples to illustrate the concept of independence in probability:

  • If P(A) = 0.3 and P(B) = 0.7, and events A and B are independent, what is P(A ∩ B)?
  • If P(A) = 0.5 and P(B) = 0.3, and events A and B are independent, what is P(A or B)?
  • If P(A) = 0.2 and P(B) = 0.8, and events A and B are independent, what is P(A ∩ B)?

These examples can be used to practice calculating the probability of two events occurring together when they are independent.

Conclusion

Q: What is the difference between independent and dependent events?

A: Independent events are events where the occurrence of one event does not affect the probability of the other event. Dependent events, on the other hand, are events where the occurrence of one event affects the probability of the other event.

Q: How do I determine if two events are independent or dependent?

A: To determine if two events are independent or dependent, you need to check if the occurrence of one event affects the probability of the other event. If it does, then the events are dependent. If it doesn't, then the events are independent.

Q: What is the formula for calculating the probability of two independent events occurring together?

A: The formula for calculating the probability of two independent events occurring together is:

P(A ∩ B) = P(A) × P(B)

Q: What is the formula for calculating the probability of two events occurring together when they are dependent?

A: The formula for calculating the probability of two events occurring together when they are dependent is:

P(A ∩ B) = P(A) × P(B | A)

where P(B | A) is the probability of event B occurring given that event A has occurred.

Q: How do I calculate the probability of event A or event B occurring?

A: To calculate the probability of event A or event B occurring, you can use the formula:

P(A or B) = P(A) + P(B) - P(A ∩ B)

Q: What is the difference between the probability of two events occurring together and the probability of two events occurring separately?

A: The probability of two events occurring together is the probability of both events occurring at the same time. The probability of two events occurring separately is the probability of one event occurring and the other event not occurring.

Q: Can two events be both independent and dependent at the same time?

A: No, two events cannot be both independent and dependent at the same time. If two events are independent, then the occurrence of one event does not affect the probability of the other event. If two events are dependent, then the occurrence of one event affects the probability of the other event.

Q: What is the concept of conditional probability?

A: Conditional probability is the probability of an event occurring given that another event has occurred. It is denoted by P(A | B) and is calculated as:

P(A | B) = P(A ∩ B) / P(B)

Q: How do I calculate the conditional probability of event A given event B?

A: To calculate the conditional probability of event A given event B, you can use the formula:

P(A | B) = P(A ∩ B) / P(B)

Q: What is the concept of Bayes' theorem?

A: Bayes' theorem is a formula for updating the probability of a hypothesis based on new evidence. It is used in Bayesian statistics and is denoted by:

P(H | E) = P(E | H) × P(H) / P(E)

where P(H | E) is the probability of the hypothesis given the evidence, P(E | H) is the probability of the evidence given the hypothesis, P(H) is the prior probability of the hypothesis, and P(E) is the probability of the evidence.

Q: How do I apply Bayes' theorem in a real-world scenario?

A: To apply Bayes' theorem in a real-world scenario, you need to identify the hypothesis, the evidence, and the prior probability of the hypothesis. You then use the formula to update the probability of the hypothesis based on the new evidence.

Conclusion

In conclusion, this article has provided answers to frequently asked questions about probability and independence. We have seen how to calculate the probability of two events occurring together when they are independent, and how to use this concept to solve problems. We have also seen some additional examples to illustrate the concept of independence in probability.