For any two events A and B in sample space S, which of the following options is NOT true?
P(A | B) = P(B | A)
(\( P(A|S) = P(A) \)) is true because conditioning on the entire sample space \( S \) does not change the probability of \( A \).
\[ P(A|S) = \frac{P(A \cap S)}{P(S)} = \frac{P(A)}{1} = P(A) \]
(\( P(A|B) = P(B|A) \)) is false in general. This equality only holds if \( P(A) = P(B) \).
\[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B|A) = \frac{P(A \cap B)}{P(A)} \]
(\( A \) and \( \emptyset \) are independent) is true because \( P(A \cap \emptyset) = P(A)P(\emptyset) = 0 \).
(\( A \) and \( S \) are independent) is true because \( P(A \cap S) = P(A) = P(A)P(S) \).
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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
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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:
