Math Problem Statement
Select the most appropriate response.
An event A will occur with probability 0.4. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.20. We may conclude
Question 9 options:
that events A and B are independent.
that events A and B are dependent.
that either A or B always occurs.
that events A and B are mutually exclusive.
Solution
To determine the correct conclusion, we can check whether events A and B are independent. For two events to be independent, the probability that both occur must equal the product of their individual probabilities, i.e.,
Let's verify this using the given information:
Calculate the product of and :
Since , which is not equal to , events A and B are not independent. This means we can rule out the first option.
Next, events are mutually exclusive if they cannot occur together, which means would be zero. Since , events A and B are not mutually exclusive, so we can rule out the last option as well.
There is no indication that either A or B always occurs, as both have probabilities less than 1, so the third option can also be ruled out.
Thus, the most appropriate response is:
Events A and B are dependent.
Would you like further clarification or details? Here are some related questions for deeper understanding:
- What is the formula for the probability of the union of two events?
- How does the concept of conditional probability relate to dependent events?
- Can you have events that are both independent and mutually exclusive?
- How can we calculate the probability of either event A or B occurring?
- What real-world examples can represent dependent events?
Tip: When dealing with probabilities, always check whether the given values satisfy the conditions for independence.
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Math Problem Analysis
Mathematical Concepts
Probability
Independent Events
Dependent Events
Mutually Exclusive Events
Formulas
P(A ∩ B) = P(A) × P(B)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Theorems
Theorem of Independent Events
Suitable Grade Level
Grades 10-12
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