Question

A fair die is rolled 12 times. Consider the following three possible outcomes:

(i) 2 2 2 2 2 2 2 2 2 2 2 2
(ii) 1 1 2 2 3 3 4 4 5 5 6 6
(iii) 4 6 2 1 3 5 2 6 4 3 1 5
Which of the following is true?
A It is absolutely impossible to get sequence (i).
B (ii) is more likely than (i).
C (iii) is more likely than (i) or (ii).
D The three outcomes are equally likely.
E Both (B) and (C) are true.

Answer 1

To determine which outcome is more likely, we need to consider the probability of each outcome occurring.

(i) The sequence (2 2 2 2 2 2 2 2 2 2 2 2) consists only of the number 2. Since each roll of the fair die has 6 possible outcomes (numbers 1 to 6), the probability of getting a sequence consisting only of 2s is (1/6)^12, which is extremely low but not absolutely impossible.

(ii) The sequence (1 1 2 2 3 3 4 4 5 5 6 6) consists of two of each number from 1 to 6. There are 12!/(2!2!2!2!2!2!) possible arrangements of these numbers, which is much larger than the probability of getting sequence (i).

(iii) The sequence (4 6 2 1 3 5 2 6 4 3 1 5) is a random arrangement of the numbers 1 to 6. Similarly to (ii), there are 12!/(2!2!2!2!2!2!) possible arrangements.

Based on these considerations, we can conclude that (ii) and (iii) are both more likely to occur than (i). Therefore, the correct answer is option E: Both (B) and (C) are true.