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# Probability

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4 12-sided dice are rolled. What is the probability that the number of dice showing a two digit number is equal to 2? Express your answer as a common fraction. (Assume that the numbers on the 12 sides are the numbers from 1 to 12 expressed in decimal.)

Apr 28, 2022

#1
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3 numbers have 2 digits (10, 11, 12)

The probability of getting a 2 digit number is $${1 \over 4}$$

We need 2 rolls to satisfy this condition, so the probability of this is $${1 \over 4} ^2$$

We also need 2 rolls to not have 2 digits. The probability of this is $${3 \over 4}^2$$

There are $$4 \choose 2$$ ways to choose which rolls are $$\ge 10$$.

Thus, the probability is:  $$\large{{1 \over 4} \times {1 \over 4} \times {3 \over 4} \times { 3 \over 4} \times {4 \choose 2}} = \color{brown}\boxed{7 \over 128}$$

Apr 28, 2022

#1
+1

3 numbers have 2 digits (10, 11, 12)

The probability of getting a 2 digit number is $${1 \over 4}$$

We need 2 rolls to satisfy this condition, so the probability of this is $${1 \over 4} ^2$$

We also need 2 rolls to not have 2 digits. The probability of this is $${3 \over 4}^2$$

There are $$4 \choose 2$$ ways to choose which rolls are $$\ge 10$$.

Thus, the probability is:  $$\large{{1 \over 4} \times {1 \over 4} \times {3 \over 4} \times { 3 \over 4} \times {4 \choose 2}} = \color{brown}\boxed{7 \over 128}$$

BuilderBoi Apr 28, 2022