A biased six-sided die is constructed such that the probability of rolling a '6' is three times the probability of rolling any other specific number (1, 2, 3, 4, or 5). The other five numbers are equally likely to occur. If this die is rolled 3 times, then the probability of getting a '6' exactly once is

Step-by-step solution

Let the probability of rolling a 1, 2, 3, 4, or 5 be p . Then, the probability of rolling a '6' is P(6) = 3p . Since the sum of the probabilities of all exhaustive elementary events is 1, we have: 5p + 3p = 1 8p = 1 p = 1 8 Thus, the probability of getting a '6' in a single roll is P(6) = 3 8 , and the probability of not getting a '6' is 1 - 3 8 = 5 8 . The die is rolled 3 times. We need the probability of getting exactly one '6'. Using the binomial probability distribution: P( exactly one 6 ) = ³C₁ ( 3 8 )¹ ( 5 8