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Questions and Answers
What does it mean when two events are mutually exclusive?
What does it mean when two events are mutually exclusive?
- They are completely independent of each other.
- They can happen at the same time.
- Their probabilities always add up to 1.
- The occurrence of one event prevents the other from happening. (correct)
How do we calculate the probability of two independent events occurring together?
How do we calculate the probability of two independent events occurring together?
- P(A) × P(B) (correct)
- P(A) + P(B)
- P(A ∩ B)
- P(A ∪ B)
What happens to the probability of an event in dependent events?
What happens to the probability of an event in dependent events?
- It becomes zero.
- It remains constant.
- It is always equal to P(A) × P(B)
- It is influenced by the occurrence of another event. (correct)
If two events are mutually exclusive, can they also be independent? Why or why not?
If two events are mutually exclusive, can they also be independent? Why or why not?
Consider the following scenario: You draw a card from a standard deck. Let A = 'drawing a red card' and B = 'drawing a heart.' Are A and B dependent or independent, and why?
Consider the following scenario: You draw a card from a standard deck. Let A = 'drawing a red card' and B = 'drawing a heart.' Are A and B dependent or independent, and why?
Why do we multiply probabilities for independent events, but use conditional probabilities for dependent events?
Why do we multiply probabilities for independent events, but use conditional probabilities for dependent events?
In a scenario where P(A) = 0.6, P(B) = 0.3, and P(A ∩ B) = 0.18, are the events A and B independent? Justify your answer.
In a scenario where P(A) = 0.6, P(B) = 0.3, and P(A ∩ B) = 0.18, are the events A and B independent? Justify your answer.
Why is it incorrect to simply add probabilities for events that are not mutually exclusive?
Why is it incorrect to simply add probabilities for events that are not mutually exclusive?
If P(A) = 0.7 and P(B) = 0.5, but P(A ∩ B) = 0.4, are A and B independent or dependent? Explain why.
If P(A) = 0.7 and P(B) = 0.5, but P(A ∩ B) = 0.4, are A and B independent or dependent? Explain why.
Why is P(B|A) = P(B) true for independent events?
Why is P(B|A) = P(B) true for independent events?
If P(A ∩ B) = 0, can A and B still be dependent? Why or why not?
If P(A ∩ B) = 0, can A and B still be dependent? Why or why not?
A student has a 70% chance of passing Math and a 60% chance of passing English. If the probability of passing both is 50%, are the events dependent?
A student has a 70% chance of passing Math and a 60% chance of passing English. If the probability of passing both is 50%, are the events dependent?
When two events are independent, why is their joint probability always smaller than or equal to the smaller individual probability?
When two events are independent, why is their joint probability always smaller than or equal to the smaller individual probability?
If P(A Ï… Î’) = 0.8 and P(A) = 0.5, P(B) = 0.4, what is P(A n B)?
If P(A Ï… Î’) = 0.8 and P(A) = 0.5, P(B) = 0.4, what is P(A n B)?
If two events are dependent, what does the conditional probability P(B|A) represent?
If two events are dependent, what does the conditional probability P(B|A) represent?
Why do mutually exclusive events have no overlap in their probabilities?
Why do mutually exclusive events have no overlap in their probabilities?
If P(A ∩ B) = 0 and P(A u B) = 0.6, what can you conclude about A and B?
If P(A ∩ B) = 0 and P(A u B) = 0.6, what can you conclude about A and B?
In a scenario where P(A) = 0.4, P(B) = 0.5, and P(A u B) = 0.7, what is P(A ∩ B)?
In a scenario where P(A) = 0.4, P(B) = 0.5, and P(A u B) = 0.7, what is P(A ∩ B)?
Why do we subtract P(A ∩ B) when calculating P(A ∪ B)?
Why do we subtract P(A ∩ B) when calculating P(A ∪ B)?
If P(A|B) = 0.5 and P(B) = 0.2, what is P(A ∩ B)?
If P(A|B) = 0.5 and P(B) = 0.2, what is P(A ∩ B)?
Why does conditional probability depend on the reduced sample space?
Why does conditional probability depend on the reduced sample space?
If A and B are dependent events and P(A) = 0.6, P(B|A) = 0.4, what is P(A ∩ B)?
If A and B are dependent events and P(A) = 0.6, P(B|A) = 0.4, what is P(A ∩ B)?
Flashcards
Mutually Exclusive Events
Mutually Exclusive Events
Events that cannot occur at the same time. Their intersection (both happening) has a probability of zero.
Independent Events
Independent Events
Events where the occurrence of one does not affect the probability of the other.
Dependent Events
Dependent Events
Events where the occurrence of one affects the probability of the other.
Joint Probability
Joint Probability
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Probability of Independent Events
Probability of Independent Events
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Conditional Probability
Conditional Probability
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P(A ∩ B)
P(A ∩ B)
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Independent Events Formula
Independent Events Formula
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Conditional Probability Formula
Conditional Probability Formula
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P(A ∪ B)
P(A ∪ B)
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Probability Addition Rule (Union)
Probability Addition Rule (Union)
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Checking Independence
Checking Independence
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Probability of Mutually Exclusive Events (Union)
Probability of Mutually Exclusive Events (Union)
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Independence vs. Dependence
Independence vs. Dependence
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Conditional Probability Equation
Conditional Probability Equation
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Study Notes
Mutually Exclusive Events
- Mutually exclusive events cannot occur together.
- Their intersection (both events happening at the same time) is zero.
- P(A ∩ B) = 0
Independent Events
- The probability of one event happening does not affect the probability of the other event happening.
- The joint probability of independent events is the product of their individual probabilities:
- P(A∩B) = P(A) x P(B)
Dependent Events
- The probability of one event happening is affected by whether or not the other event has happened.
- The probability of one event given the other event has occurred is calculated using conditional probability:
- P(B|A) = P(A ∩ B)/P(A)
- P(A ∩ B) is the joint probability of both events happening.
Conditional Probability
- Conditional probability is the probability of an event (B) occurring given that another event (A) has already occurred.
- P(B|A)
Calculating Joint probability
- When calculating the joint probability of two events (P(A and B) or P(A∩B)) consider if the events are independent or dependent, and if they are mutually exclusive. If they are not mutually exclusive, to find the probability of both occurring, calculate the conditional probability and multiply by the probability of the preceding event.
- To find P(A ∪ B) (the probability of either event occurring), use the formula P(A) + P(B) – P(A ∩ B) (for non-mutually exclusive events).
Example calculation of Probability
- Example:
- P(A) = 0.6
- P(B) = 0.3
- P(A∩B) = P(A) x P(B) = 0.18
- The calculation shows that P(A∩B) = P(A) x P(B). Therefore, A and B are independent events as the probability of event B occurring doesn't' change after event A occurs.
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Description
Test your understanding of mutually exclusive, independent, and dependent events with this quiz. Explore the concepts of joint and conditional probability through practical examples and calculations. Enhance your grasp of probability theory in this informative assessment.
