Let A, B, C and D be mutually exclusive and exhaustive events and $\frac{P(A)}{2} = \frac{P(B)}{3} = \frac{P(C)}{5} = \frac{P(D)}{8}$.
To solve the problem, we are given that the events \(A\), \(B\), \(C\), and \(D\) are mutually exclusive and exhaustive. Furthermore, it is given that:
\(\frac{P(A)}{2} = \frac{P(B)}{3} = \frac{P(C)}{5} = \frac{P(D)}{8} = k\)
Here, \(k\) is a constant. From this equation, we can express the probabilities of each event as:
- \(P(A) = 2k\)
- \(P(B) = 3k\)
- \(P(C) = 5k\)
- \(P(D) = 8k\)
Since the events are exhaustive, their probabilities sum to 1:
\(P(A) + P(B) + P(C) + P(D) = 1\)
Substituting the expressions for the probabilities, we have:
\(2k + 3k + 5k + 8k = 1\)
Simplifying this gives:
\(18k = 1\)
Therefore, \(k = \frac{1}{18}\).
Next, we need to find the value of:
\(\frac{[2P(A) + 3P(B)]}{[4P(C) + 5P(D)]}\)
Substituting the values of the probabilities:
\(\text{Numerator} = 2(2k) + 3(3k) = 4k + 9k = 13k\)
\(\text{Denominator} = 4(5k) + 5(8k) = 20k + 40k = 60k\)
Thus, the expression simplifies to:
\(\frac{13k}{60k} = \frac{13}{60}\)
The correct answer is therefore \(\frac{13}{60}\).
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Important Questions from Probability
Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?
If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?
A. 2/3
B. 3/4
C. 1/4
D. 1/9Statements followed by some conclusions are given below.
Statements:
1. A bag has 2 white, 3 black, 4 red and 6 green balls.
2. 1 ball selected at random from the bag.
Conclusions:
I. The probability that a black ball is selected is 1/5
II. The probability that a red ball is selected is 6/15
Find which of the conclusions logically follows from the given statement
A. Only conclusion I follows.
B. Only conclusion II follows.
C. Both I and II follow.
D. Neither I nor II follows.
In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?
A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
