Understanding Probability Concepts for Events A, B, C
This problem involves understanding the properties of mutually exclusive and exhaustive events in probability.
- Mutually Exclusive Events: These are events that cannot occur at the same time. If events A, B, and C are mutually exclusive, it means that the occurrence of one event precludes the occurrence of the others. Mathematically, \(P(A \cap B) = 0\), \(P(A \cap C) = 0\), and \(P(B \cap C) = 0\).
- Exhaustive Events: These are events that cover all possible outcomes of an experiment. If events A, B, and C are exhaustive, their union represents the entire sample space, meaning \(P(A \cup B \cup C) = 1\).
For mutually exclusive and exhaustive events, the sum of their individual probabilities is equal to 1:
\( P(A) + P(B) + P(C) = 1 \)
Analyzing Given Probability Relationships
The question provides the following relationships between the probabilities of events A, B, and C:
- \(3P(B) = 4P(A)\)
- \(3P(C) = 2P(B)\)
From the first relationship, we can express \(P(B)\) in terms of \(P(A)\): \( P(B) = \frac{4}{3} P(A) \)
From the second relationship, we can express \(P(C)\) in terms of \(P(B)\): \( P(C) = \frac{2}{3} P(B) \)
Calculating P(A) Step-by-Step
Now, let's substitute the expression for \(P(B)\) into the equation for \(P(C)\) to get \(P(C)\) in terms of \(P(A)\): \( P(C) = \frac{2}{3} \left( \frac{4}{3} P(A) \right) \) \( P(C) = \frac{8}{9} P(A) \)
We know that for these mutually exclusive and exhaustive events, the sum of their probabilities is 1:
\( P(A) + P(B) + P(C) = 1 \)
Substitute the expressions for \(P(B)\) and \(P(C)\) in terms of \(P(A)\) into this equation:
\( P(A) + \frac{4}{3} P(A) + \frac{8}{9} P(A) = 1 \)
To solve for \(P(A)\), find a common denominator, which is 9:
\( \frac{9}{9} P(A) + \frac{12}{9} P(A) + \frac{8}{9} P(A) = 1 \)
Combine the terms on the left side:
\( \left( \frac{9 + 12 + 8}{9} \right) P(A) = 1 \) \( \frac{29}{9} P(A) = 1 \)
Finally, isolate \(P(A)\):
\( P(A) = \frac{9}{29} \)
Therefore, the probability of event A, \(P(A)\), is 9/29.
Similar Questions
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Important Questions from Probability
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
The probability of having 53 Tuesdays in an ordinary year is:
When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
