Probability

Multiples of 5: 70/5 = 14. Multiples of 7: 70/7 = 10. Multiples of both (35): 70/35 = 2. Total favorable = 14 + 10 - 2 = 22. Probability = 22/70 = 11/35.

Multiples of 3 on a die are 3 and 6. For two dice, the favorable outcomes are (3,3), (3,6), (6,3), and (6,6), totaling 4 outcomes. The probability is 4/36, which simplifies to 1/9.

Prime numbers less than 50 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (15 total). Primes less than 24 are: 2, 3, 5, 7, 11, 13, 17, 19, 23 (9 total). Probability = 9/15 = 3/5.

Bag A: P(Green) = 4/10 = 2/5. Bag B: P(Green) = 3/7. Since events are independent, P(Both Green) = (2/5) * (3/7) = 6/35.

The probability of at least one tail is 1 minus the probability of getting no tails (all heads). The probability of all heads in 3 tosses is (1/2)^3 = 1/8. Thus, 1 - 1/8 = 7/8.

A standard die has 6 faces (1, 2, 3, 4, 5, 6). Numbers greater than 2 are 3, 4, 5, 6, which are 4 outcomes. The probability is 4/6, which simplifies to 2/3.

Probability of head = 1/2. Probability of odd number (1, 3, 5) = 3/6 = 1/2. Since events are independent, P(Head and Odd) = 1/2 * 1/2 = 1/4.

There are 4 kings in a standard deck of 52 cards. The probability of drawing one is 4/52.

There are 12 face cards (Jack, Queen, King of 4 suits) in a 52-card deck. Probability = 12/52 = 3/13.

There are 2 black kings (King of Spades and King of Clubs) in a deck of 52 cards. Probability = 2 / 52 = 1 / 26.

There are 13 spades and 4 aces, but the ace of spades is counted twice. Thus, favourable cards = 13 + 4 - 1 = 16, giving probability 16/52 = 4/13.

Numbers are 2 to 101 (100 total). Perfect cubes are 8 (2^3), 27 (3^3), 64 (4^3). There are 3 such numbers. Probability = 3/100.

A standard die has outcomes from 1 through 6, so obtaining 7 is impossible. The probability is therefore 0, making option D correct.

A standard die has 6 faces: {1, 2, 3, 4, 5, 6}. The even numbers are {2, 4, 6}, which are 3 outcomes out of 6 total outcomes, resulting in a probability of 3/6 = 1/2.

Total tosses are 200 and heads appeared 120 times, so tails appeared 200 - 120 = 80 times. The probability of getting a tail is 80/200, which simplifies to 8/20 or 2/5.