The probability of rolling a six on a biased dice is five-sixths. If the dice is rolled twice, what is the probability of not getting a six on either roll? Circle your answer the options are two twelfths, one over 36, 25 over 36, or ten twelfths.
We’re told that this dice is biased, which just means that there isn’t an equal probability of it landing on each face. It’s been weighted in some way so that the probability of it landing on a six is five- sixths. We’re told that the dice is going to be rolled twice, and we are asked for the probability that we don’t get a six on either roll. So that means the first roll isn’t a six and the second roll isn’t a six either.
Before we can work out the probability of not getting a six on either roll, we must first work out the probability of not getting a six on each individual roll. To do so, we can subtract the probability that we do get a six from one because the events of getting a six or not getting a six are what’s known as complementary events; it’s one or the other. So their probabilities need to sum to one. We’re given in the question that the probability of rolling a six on this dice is five-sixths. So we have one minus five six which is equal to one-sixth.
So now we know the probability that we don’t get a six on each roll, but what about the probability that we don’t get a six on either roll when we roll the dice twice? Well we can use the fact that if two events 𝐴 and 𝐵 are independent, then if we want to find the probability of them both happening, we can multiply their individual probabilities together. To find the probability then that first roll is not a six and the second roll is not a six, we multiply these individual probabilities, both of which one-sixth, together. We can do this because successive rolls of a dice are independent.
This dice is biased meaning it doesn’t have an equal probability of landing on each of its faces. But whatever the outcome of the first roll is, this won’t affect the probabilities for the second. So the outcomes of the two rolls of independent. To multiply fractions together, we multiply the numerators and then multiply the denominators. So we have one over 36. We’ve found our answer, but let’s have a look at some of the other possible options we were given because they may show us some common misconceptions.
The answer of 25 over 36 could have been found by multiplying five-sixths by five-sixths, but this would give the probability of rolling a six both times rather than rolling a six neither time. The answer two twelfths is perhaps an attempt at summing the individual probabilities together: one-sixth plus one sixth. But a mistake has been made here because the fractions have been added incorrectly. The two denominators have been added as well as the numerators. The answer of ten twelfths is again perhaps an attempt to adding two probabilities together. But this time, they’re the probabilities that we do roll six on each roll. But again, a mistake has been made in the addition of the two fractions. The correct answer to this question, found by multiplying the probability of not getting a six on each roll together, is one over 36.