91Ó°ÊÓ

\(\mathrm{P}(\mathrm{A})=0.3, \mathrm{P}(\mathrm{B})=0.7\) (a) Can you compute \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})\) if you only know \(\mathrm{P}(\mathrm{A})\) and \(\mathrm{P}(\mathrm{B}) ?\) (b) Assuming that events \(A\) and \(B\) arise from independent random processes, i. what is \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})\) ? ii. what is \(\mathrm{P}(\mathrm{A}\) or \(\mathrm{B}) ?\) iii. what is \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) ?\) (c) If we are given that \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})=0.1,\) are the random variables giving rise to events \(\mathrm{A}\) and \(\mathrm{B}\) independent? (d) If we are given that \(\mathrm{P}(\mathrm{A}\) and \(\mathrm{B})=0.1,\) what is \(\mathrm{P}(\mathrm{A} \mid \mathrm{B}) ?\)

Step by step solution

01

Understanding given probabilities

We know \( \mathrm{P}(\mathrm{A}) = 0.3 \) and \( \mathrm{P}(\mathrm{B}) = 0.7 \). These represent the probabilities of events A and B occurring individually.

02

Solving part (a) - Joint Probability

No, you cannot compute \( \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) \) solely from \( \mathrm{P}(\mathrm{A}) \) and \( \mathrm{P}(\mathrm{B}) \) unless you know whether A and B are independent.

03

Understanding independence

If events A and B are independent, the probability of both events occurring is the product of their probabilities: \( \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = \mathrm{P}(\mathrm{A}) \times \mathrm{P}(\mathrm{B}) \).

04

Solving part (b)i - Joint Probability with independence

Assuming independence, calculate \( \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = 0.3 \times 0.7 = 0.21 \).

05

Solving part (b)ii - Union Probability with independence

Using the formula for the union of two independent events: \( \mathrm{P}(\mathrm{A} \text{ or } \mathrm{B}) = \mathrm{P}(\mathrm{A}) + \mathrm{P}(\mathrm{B}) - \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = 0.3 + 0.7 - 0.21 = 0.79 \).

06

Solving part (b)iii - Conditional Probability with independence

For independent events, \( \mathrm{P}(\mathrm{A} \mid \mathrm{B}) = \mathrm{P}(\mathrm{A}) = 0.3 \) because B occurring does not affect the probability of A.

07

Analyzing part (c) - Independence analysis with given joint probability

We are given \( \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = 0.1 \). Since \( 0.1 eq 0.21 \), A and B are not independent.

08

Solving part (d) - Conditional Probability with given joint probability

Given \( \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = 0.1 \), use the formula: \( \mathrm{P}(\mathrm{A} \mid \mathrm{B}) = \frac{\mathrm{P}(\mathrm{A} \text{ and } \mathrm{B})}{\mathrm{P}(\mathrm{B})} = \frac{0.1}{0.7} \approx 0.143 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Independent Events

In probability theory, events are termed independent if the occurrence of one event does not influence the occurrence of another.
This means the probability of both events happening together (joint probability) is the product of their individual probabilities.
Mathematically, for two independent events A and B, the formula is: \[ \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = \mathrm{P}(\mathrm{A}) \times \mathrm{P}(\mathrm{B}) \]

• For example, if the probability of event A is 0.3 and event B is 0.7, and they are independent, then the probability of both A and B occurring is 0.21 (0.3 multiplied by 0.7).
• Independence implies that knowing whether one event occurs does not update or change the probability of the other event in any way.

Remember, if a given joint probability deviates from this product, it indicates a lack of independence.

Exploring Conditional Probability

Conditional probability helps us understand the likelihood of an event occurring given that another event has already occurred. This concept is crucial when events are dependent.
To find the conditional probability of event A given event B has occurred, we use the formula: \[ \mathrm{P}(\mathrm{A} \mid \mathrm{B}) = \frac{\mathrm{P}(\mathrm{A} \text{ and } \mathrm{B})}{\mathrm{P}(\mathrm{B})} \]

• This formula tells us how much information or change in probability occurs once event B is known to happen.
• For instance, if \( \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) = 0.1 \) and \( \mathrm{P}(\mathrm{B}) = 0.7 \), then \( \mathrm{P}(\mathrm{A} \mid \mathrm{B}) \) is approximately 0.143.

This allows you to rethink probabilities in light of new evidence or occurrences.

Joint Probability Investigated

Joint probability refers to the probability of multiple events occurring together.
When dealing with joint probability, it's important to know whether events are independent or dependent.

• For independent events, joint probability is simply \( \mathrm{P}(\mathrm{A}) \times \mathrm{P}(\mathrm{B}) \).
• If events are not independent, like when given the joint probability as \( 0.1 \), it prompts us to analyze the impact of combined outcomes beyond simple multiplication.

Understanding joint probability can provide deeper insight into how combinations of events behave in real-world scenarios.

Union of Events Simplified

The union of two events, often expressed as "A or B," refers to any occurrence of either event.
This involves calculating the probability that at least one of the events occurs.
The union's probability is given by: \[ \mathrm{P}(\mathrm{A} \text{ or } \mathrm{B}) = \mathrm{P}(\mathrm{A}) + \mathrm{P}(\mathrm{B}) - \mathrm{P}(\mathrm{A} \text{ and } \mathrm{B}) \]

• This formula makes sure that the overlap is not counted twice, especially when events A and B are not mutually exclusive.
• In our example, if \( \mathrm{P}(\mathrm{A}) = 0.3 \) and \( \mathrm{P}(\mathrm{B}) = 0.7 \) with their joint probability being \( 0.21 \), their union is \( 0.79 \).

Remember, the union accounts for scenarios when at least one event occurs, painting a broader picture of probability outcomes.