High School

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------------------------------------------------ 80. Two dice are thrown simultaneously. Find the probability that the sum of the digits on the two dice is 8 or more.

(A) \(\frac{5}{18}\)
(B) \(\frac{5}{12}\)
(C) \(\frac{5}{36}\)
(D) \(\frac{7}{12}\)

81. A number is selected from the first 20 natural numbers. Find the probability that it is divisible by 3 or 7.

(A) \(\frac{7}{20}\)
(B) \(\frac{12}{37}\)
(C) \(\frac{24}{67}\)
(D) \(\frac{18}{20}\)

Answer :

Let's consider each question step by step.

Question 1: Probability of Sum of Digits on Dice Being 8 or More

When two dice are thrown, each die can show a number from 1 to 6. The total number of possible outcomes is:

[tex]6 \times 6 = 36.[/tex]

We need to find the probability that the sum of the digits on the dice is 8 or more. Let's list the possible outcomes where the sum is 8 or more:


  • Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2)

  • Sum of 9: (3,6), (4,5), (5,4), (6,3)

  • Sum of 10: (4,6), (5,5), (6,4)

  • Sum of 11: (5,6), (6,5)

  • Sum of 12: (6,6)


Counting these outcomes gives us a total of 15 favorable outcomes.

The probability is calculated as:

[tex]\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{15}{36} = \frac{5}{12}.[/tex]

Therefore, the correct option is (B) [tex]\frac{5}{12}[/tex].

Question 2: Probability of a Number Being Divisible by 3 or 7

We have numbers from 1 to 20. We need to find how many of these numbers are divisible by 3 or 7.


  • Numbers divisible by 3 within 1 to 20: 3, 6, 9, 12, 15, 18. That's 6 numbers.

  • Numbers divisible by 7 within 1 to 20: 7, 14. That's 2 numbers.


To avoid double counting, we check if any number is divisible by both 3 and 7. That would mean divisible by 21, which isn't within the range 1 to 20.

Therefore, the total number of unique favorable outcomes is [tex]6 + 2 = 8[/tex].

The probability is calculated as:

[tex]\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{8}{20} = \frac{2}{5}.[/tex]

However, this option is not among the given choices. Rechecking the problem statement for any discrepancy or error might be necessary. Assuming no error, none of the given options match, indicating a potential mistake in the question part or set of solutions provided.