Probability that at least one...problem (Manhattan Ch.23 #29)

GRE Quant Problem Solving
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The answer explanation says to use the 1-x shortcut, meaning:

  1. Get the probability of getting a number less than or equal to 4 in both rolls = 4/6 x 4/6 = 4/9
  2. Subtract above from 1 = 1-4/9 = 5/9

My question is: Why do we subtract from 1? Does this mean the probability that at least one of the rolls resulting in a number greater than 4 basically means all the possibilities of getting a number greater than 4 out of both rolls?

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this is because the sum of the probability of the event happening and the probability of the event not happening is equal to 1.

What this means is that when you roll the die twice you must satisfy one of the conditions:

  1. greater than 4 on the first die and any other number on the second die
  2. any other number on the first die and greater than 4 on the second die
  3. greater than 4 on the first die and greater than 4 on the second die

the probability for the 1st condition is \cfrac{2}{6} \times \cfrac{6}{6} = \cfrac{1}{3}
the probability for the 2nd condition is \cfrac{6}{6} \times \cfrac{2}{6} = \cfrac{1}{3}
the probability for the 3rd condition is \cfrac{2}{6} \times \cfrac{2}{6} = \cfrac{1}{9}

using the formula

P(A \cup B) = P(A) + P(B) - P(A \cap B)

therefore the probability = \cfrac{1}{3} + \cfrac{1}{3} - \cfrac{1}{9} = \cfrac{5}{9}
(\cfrac{1}{9} is substracted because it is the both case and it is double counted)

that’s why it is better to use the 1 - x shortcut.

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3

Thank you!