Multiple Choice Exam Setup
The problem involves 30 multiple-choice questions, each with 4 options. Three students (A, B, C) answer randomly and independently. We need the probability that all their answers match perfectly across all 30 questions.
Single Question Match Probability
For any single question:
- There are 4 possible options.
- Student A picks one option randomly.
- The probability that Student B picks the same option as A is $1/4$.
- The probability that Student C picks the same option as A is $1/4$.
- Since B and C choose independently, the probability that both B and C match A's choice is: $P(\text{match}) = P(\text{B matches A}) \times P(\text{C matches A}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$
30 Question Match Probability
There are 30 independent questions.
The probability of all 30 answers matching is the probability of a single question match raised to the power of 30:
$ P(\text{all 30 match}) = (P(\text{match}))^{30} = \left(\frac{1}{16}\right)^{30} $
Expressing 16 as $4^2$ and using exponent rules:
$ \left(\frac{1}{16}\right)^{30} = \left(\frac{1}{4^2}\right)^{30} = (4^{-2})^{30} = 4^{-2 \times 30} = 4^{-60} $
The final probability is $4^{-60}$.
Important Questions from Discrete Probability
- Suppose two fair dice are thrown independently at random. Let $X$ and $Y$ be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?
- A box contains 40 numbered red balls and 60 numbered black balls. From the box, balls are drawn one by one at random without replacement till all the balls are drawn. The probability that the last ball drawn is black equals
Ten balls are put in 6 slots at random. Then the expected total number of balls in the two extreme slots is
- There are five empty boxes. Balls are placed independently one after another in randomly selected boxes. The probability that the fourth ball is the first to be placed in an occupied box equals
Suppose $A, B, C$ are events in a common probability space with
$P(A) = 0.2, \ P(B) = 0.2, \ P(C) = 0.3$, $P(A \cap B) = 0.1, \ P(A \cap C) = 0.1, \ P(B \cap C) = 0.1$.
Which of the following are possible values of $P(A \cup B \cup C)$?
