1. Problem: Find the probability of getting an even number when a die is rolled. The possible even numbers on a die are 2, 4, and 6. So, there are 3 favorable outcomes. Total outcomes = 6. Probability = number of favorable outcomes / total outcomes = $\frac{3}{6} = \frac{1}{2}$. 2. Problem: Find the probability that the sum of two dice rolled is 7. Possible pairs that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 pairs. Total outcomes = $6 \times 6 = 36$. Probability = $\frac{6}{36} = \frac{1}{6}$. 3. Problem: Find the probability of getting heads when a fair coin is tossed once. There are 2 possible outcomes: Heads and Tails. Probability = $\frac{1}{2}$. 4. Problem: The probability of rain tomorrow is 0.3. Find the probability it will not rain. Using complementary probability: Probability(not rain) = $1 - 0.3 = 0.7$. 5. Problem: A bag contains 3 red balls and 2 blue balls. Find the probability that a randomly drawn ball is not red. Total balls = 3 + 2 = 5. Number of non-red balls = 2. Probability = $\frac{2}{5}$. 6. Problem: A card is drawn from a standard deck. Probability it is not a face card. Face cards = Jack, Queen, King in each of 4 suits = 12 cards. Total cards = 52. Probability(not face) = $\frac{52 - 12}{52} = \frac{40}{52} = \frac{10}{13}$. 7. Problem: Lottery ticket costs 1, prize is 10 with probability 0.1. Find expected value. Expected winning = $10 \times 0.1 = 1$. Expected cost = 1. Expected value = Expected winning - cost = $1 - 1 = 0$. 8. Problem: A player pays 5 to play; wins 30 if rolls 6, else wins 0. Find expected net gain/loss. Probability(roll 6) = $\frac{1}{6}$. Expected winnings = $30 \times \frac{1}{6} + 0 \times \frac{5}{6} = 5$. Net gain/loss = Expected winnings - cost = $5 - 5 = 0$. 9. Problem: Probability of winning is 0.25. Find odds in favor. Odds in favor = $\frac{P(Win)}{1 - P(Win)} = \frac{0.25}{0.75} = \frac{1}{3}$. 10. Problem: Odds in favor of drawing a heart from a deck. Number of hearts = 13. Number of non-hearts = 52 - 13 = 39. Odds in favor = $13:39 = 1:3$. 11. Problem: Odds against rolling a prime number on a die. Prime numbers on a die: 2,3,5 = 3. Non-prime = 6 - 3 = 3. Odds against = number of unfavorable : favorable = $3:3 = 1:1$. 12. Problem: Odds in favor of drawing two black balls in succession without replacement from 7 white and 5 black balls. Total balls = 12. First black ball probability = $\frac{5}{12}$. After drawing one black, remaining black = 4, total = 11. Second black probability = $\frac{4}{11}$. Probability both black = $\frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$. Probability not both black = $1 - \frac{5}{33} = \frac{28}{33}$. Odds in favor = $\frac{5}{33} : \frac{28}{33} = 5 : 28$. 13. Problem: Probability of getting either 2 or 5 on a die. Favorable outcomes = 2 (either 2 or 5). Total = 6. Probability = $\frac{2}{6} = \frac{1}{3}$. 14. Problem: Probability of drawing either a heart or a king from 52 cards. Hearts = 13, Kings = 4. King of hearts counted twice, so subtract 1. Total favorable = $13 + 4 - 1 = 16$. Probability = $\frac{16}{52} = \frac{4}{13}$. 15. Problem: Probability a student takes Mathematics or Physics given 60% take Mathematics, 45% take Physics, 25% take both. Use formula: $P(M \cup P) = P(M) + P(P) - P(M \cap P)$. $= 0.6 + 0.45 - 0.25 = 0.8$. Final answers: 1. $\frac{1}{2}$ 2. $\frac{1}{6}$ 3. $\frac{1}{2}$ 4. 0.7 5. $\frac{2}{5}$ 6. $\frac{10}{13}$ 7. 0 8. 0 9. $\frac{1}{3}$ 10. $1:3$ 11. $1:1$ 12. $5:28$ 13. $\frac{1}{3}$ 14. $\frac{4}{13}$ 15. 0.8