Simplify 2/x+3 - 3/x-5
1 answer:
First, you should equalize the denominator of first and second fraction. If the denominator multiplies by (x-5), so does the numerator. If the denominator multiplies by (x+3), so does the numerator.
(Look into my attachment at the second row)
Second, simplify the algebraic form for the numerator.
(Look into my attachment at the fourth row)
Third, simplify the denominator
(Look into my attachment at the last row)
(Look into my attachment at the second row)
Second, simplify the algebraic form for the numerator.
(Look into my attachment at the fourth row)
Third, simplify the denominator
(Look into my attachment at the last row)
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You can see the three diagram attached. Each link is labeled with the probability: you have probability 1/6 that a six is rolled, and 5/6 that it is not rolled.
To answer the questions, find the path that brings you to the desired outcome, and multiply all the labels you meet.
First question:
To get three sixes, you have to choose the left path at each roll. The probability is always 1/6, so the answer is
Second question:
To get no sixes, you have to choose the right path at each roll. The probability is always 5/6, so the answer is
Third question:
To get exactly one six, it can either be the first, second or third roll.
In all cases, you have to choose the left path once and the right path twice: left-right-right mean that you get the six in the first roll, right-left-right means that you get the six in the second roll, right-right-left means that you get the six in the third roll.
In every case, the left turn has probability 1/6, and the right turn has probability 5/6. The probability of each combination is thus
And since there are three of these combinations, The answer is
Fourth question:
Since the question suggests to use what we already achieved, let's do it: having at least one six is the complementary event of having no sixes at all. If an event has probability p, its complementary has probability 1-p. So, since the probability of no sixes is known, the probability of at least one six is
