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The probability of rolling a six on a biased dice is five-sixths.
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If the dice is rolled twice, what is the probability of not getting a six on either roll?
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Circle your answer the options are two twelfths, one over 36, 25 over 36, or ten twelfths.
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We’re told that this dice is biased, which just means that there isn’t an equal probability of it landing on each face.
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It’s been weighted in some way so that the probability of it landing on a six is five- sixths.
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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.
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So that means the first roll isn’t a six and the second roll isn’t a six either.
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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.
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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.
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So their probabilities need to sum to one.
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We’re given in the question that the probability of rolling a six on this dice is five-sixths.
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So we have one minus five six which is equal to one-sixth.
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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?
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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.
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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.
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We can do this because successive rolls of a dice are independent.
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This dice is biased meaning it doesn’t have an equal probability of landing on each of its faces.
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But whatever the outcome of the first roll is, this won’t affect the probabilities for the second.
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So the outcomes of the two rolls of independent.
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To multiply fractions together, we multiply the numerators and then multiply the denominators.
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So we have one over 36.
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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.
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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.
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The answer two twelfths is perhaps an attempt at summing the individual probabilities together: one-sixth plus one sixth.
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But a mistake has been made here because the fractions have been added incorrectly.
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The two denominators have been added as well as the numerators.
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The answer of ten twelfths is again perhaps an attempt to adding two probabilities together.
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But this time, they’re the probabilities that we do roll six on each roll.
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But again, a mistake has been made in the addition of the two fractions.
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The correct answer to this question, found by multiplying the probability of not getting a six on each roll together, is one over 36.