Two dice are thrown at the same time and the product of the numbers appearing on them is noted. The probability that the product of the numbers lies between 8 and 13 is

Step 1: Determine the sample space

When two dice are thrown, each die has 6 possible outcomes (1, 2, 3, 4, 5, 6). The total number of possible outcomes in the sample space is the product of the outcomes for each die.

Step 2: List outcomes with product between 8 and 13

We need to find pairs of numbers whose product is greater than 8 and less than 13. We list all such pairs: $(2{,}4)$ product is 8 (not included), $(2{,}5)$ product is 10, $(2{,}6)$ product is 12, $(3{,}3)$ product is 9, $(3{,}4)$ product is 12, $(4{,}2)$ product is 8 (not included), $(4{,}3)$ product is 12, $(5{,}2)$ product is 10, $(6{,}2)$ product is 12. The products must be strictly between 8 and 13, so 8 and 13 are excluded. The favorable outcomes are those with products 9, 10, 11, 12.

Step 3: Count favorable outcomes

Counting the listed pairs from the previous step, we find there are 7 favorable outcomes.

Step 4: Calculate the probability

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

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