Probability

The only way to get a sum of 12 with two dice is (6, 6). There is 1 favorable outcome out of 36 total outcomes.

Multiples of 5 between 1 and 20 are 5, 10, 15, 20. There are 4 such numbers. Probability = 4/20 = 1/5.

Primes between 1 and 17 are 2, 3, 5, 7, 11, 13, 17. There are 7 such numbers. Total numbers are 17. Probability = 7/17.

Numbers 1 to 17. Odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17. There are 9 odd numbers. Total numbers = 17. Probability = 9/17.

Numbers divisible by both 2 and 3 must be divisible by 6. In the range 1 to 17, these are 6 and 12. There are 2 such numbers. Probability = 2/17.

Total outcomes = 36. Sums divisible by 4: 4, 8, 12 (3, 5, 1 outcomes = 9). Sums divisible by 6: 6, 12 (5, 1 outcomes = 6). Sum 12 is counted twice. Total favorable = 9 + 6 - 1 = 14. Probability = 14/36 = 7/18.

There are 4 aces in a standard deck of 52 cards. Probability = 4/52 = 1/13.

Numbers greater than 4 on a die are 5 and 6. There are 2 favorable outcomes out of 6. Probability = 2/6 = 1/3.

There is only one jack of hearts in a standard deck of 52 cards. Therefore, the probability is 1/52.

Total marbles = 3 + 7 + 10 = 20. White marbles = 10. Probability = 10/20 = 1/2.

A standard die has outcomes {1, 2, 3, 4, 5, 6}. Composite numbers are numbers greater than 1 that are not prime; in this set, they are 4 and 6. The probability is 2/6, which simplifies to 1/3.

There are 5 cards in total (Ten, Queen, King, Jack, Ace). The probability of picking a Jack is the number of Jacks (1) divided by the total number of cards (5), which is 1/5.

Total balls = 7 + 10 = 17. Black balls = 10. Probability = 10/17.

A standard die has outcomes {1, 2, 3, 4, 5, 6}. The numbers lying strictly between 1 and 6 are {2, 3, 4, 5}. There are 4 such numbers, so the probability is 4/6 = 2/3.

With two standard dice, a sum of 9 occurs in four ordered outcomes: (3,6), (4,5), (5,4), and (6,3). Since there are 36 equally likely outcomes, the probability is 4/36 = 1/9.