Question

There are 21 cubic dices in a box: 14 of them are regular, and 7 are false. "False" dices have sixes on all sides. We repeat the following experiment 3 times: we draw a die (with replacement) and roll it once.

Calculate the probability that sixes will be the outcomes of all the rolls.

Answer 1

Total 21 dices - 14 regular, 7 false
Let event S denote that roll of a dice gives 6.
Let event R denote that a regular dice is selected
P(R) = 14/21 = 2/3
Let evnet F denote that a false dice is selected.
P(F) = 7/21 = 1/3
So, P(S | R) = 1/6 and P(S | F) = 1

First we calculate the probability of getting '6' on a roll of cubic dice
By total law of probability:
P(S) = P(R).P(S|R) + P(F).P(S|F)
P(S) = (2/3 x 1/6) + (1/3 x 1) = 0.444

Hence probability of success in our experiment is p = 0.444(rolling a 6)
Probability that we get '6' on all the 3 rolls = ³C₃ (0.444)³(1-0.444)⁰ = 0.0875