Q1: A fair six-sided die is rolled. What is the probability of rolling a number greater than 4? (a) \(\frac{1}{6}\) (b) \(\frac{1}{3}\) (c) \(\frac{1}{2}\) (d) \(\frac{2}{3}\)
Solution:
Ans: (b) Explanation: The numbers greater than 4 on a six-sided die are 5 and 6. There are 2 favorable outcomes out of 6 possible outcomes. Therefore, the probability is \(\frac{2}{6} = \frac{1}{3}\).
Q2: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability that it is NOT blue? (a) \(\frac{3}{10}\) (b) \(\frac{7}{10}\) (c) \(\frac{1}{2}\) (d) \(\frac{2}{5}\)
Solution:
Ans: (b) Explanation: Total marbles = 5 + 3 + 2 = 10. Non-blue marbles = 5 red + 2 green = 7. The probability of NOT drawing a blue marble is \(\frac{7}{10}\).
Q3: What is the probability of flipping a fair coin and getting heads? (a) 0 (b) \(\frac{1}{4}\) (c) \(\frac{1}{2}\) (d) 1
Solution:
Ans: (c) Explanation: A fair coin has two equally likely outcomes: heads and tails. The probability of getting heads is \(\frac{1}{2}\).
Q4: A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability of landing on an even number? (a) \(\frac{1}{4}\) (b) \(\frac{3}{8}\) (c) \(\frac{1}{2}\) (d) \(\frac{5}{8}\)
Solution:
Ans: (c) Explanation: The even numbers from 1 to 8 are 2, 4, 6, and 8. There are 4 favorable outcomes out of 8 possible outcomes. The probability is \(\frac{4}{8} = \frac{1}{2}\).
Q5: If the probability of an event occurring is \(\frac{3}{7}\), what is the probability of the event NOT occurring? (a) \(\frac{3}{7}\) (b) \(\frac{4}{7}\) (c) \(\frac{1}{7}\) (d) \(\frac{7}{3}\)
Solution:
Ans: (b) Explanation: The probability of an event not occurring is equal to 1 minus the probability of the event occurring. Therefore, \(1 - \frac{3}{7} = \frac{7}{7} - \frac{3}{7} = \frac{4}{7}\).
Q6: A deck of 52 playing cards contains 4 aces. What is the probability of drawing an ace from a well-shuffled deck? (a) \(\frac{1}{52}\) (b) \(\frac{1}{13}\) (c) \(\frac{1}{4}\) (d) \(\frac{4}{13}\)
Solution:
Ans: (b) Explanation: There are 4 aces in a deck of 52 cards. The probability of drawing an ace is \(\frac{4}{52} = \frac{1}{13}\).
Q7: Which of the following probabilities is impossible? (a) 0 (b) 0.25 (c) 0.5 (d) 1.5
Solution:
Ans: (d) Explanation: Probability values must be between 0 and 1, inclusive. A probability of 1.5 is greater than 1, which is impossible. Option (a) represents an impossible event, option (b) and (c) represent possible probabilities, but option (d) is not a valid probability value.
Q8: A jar contains 12 identical balls numbered 1 through 12. What is the probability of randomly selecting a ball with a number that is a multiple of 3? (a) \(\frac{1}{6}\) (b) \(\frac{1}{4}\) (c) \(\frac{1}{3}\) (d) \(\frac{5}{12}\)
Solution:
Ans: (c) Explanation: The multiples of 3 from 1 to 12 are 3, 6, 9, and 12. There are 4 favorable outcomes out of 12 possible outcomes. The probability is \(\frac{4}{12} = \frac{1}{3}\).
## Section B: Fill in the Blanks Q9: The probability of an event that is certain to occur is __________.
Solution:
Ans: 1 Explanation: A certain event has a probability of 1, meaning it will definitely happen.
Q10: The sum of the probabilities of all possible outcomes in a sample space equals __________.
Solution:
Ans: 1 Explanation: In any sample space, the probabilities of all possible outcomes must add up to 1, representing certainty that one of the outcomes will occur.
Q11: If an event cannot occur, its probability is __________.
Solution:
Ans: 0 Explanation: An impossible event has a probability of 0, meaning it will never happen.
Q12: The set of all possible outcomes of an experiment is called the __________.
Solution:
Ans: sample space Explanation: The sample space is the complete set of all possible outcomes that can result from a probability experiment.
Q13: In a standard deck of 52 cards, there are __________ suits.
Solution:
Ans: 4 Explanation: A standard deck contains four suits: hearts, diamonds, clubs, and spades, each containing 13 cards.
Q14: The probability of an event is calculated as the number of favorable outcomes divided by the total number of __________ outcomes.
Solution:
Ans: possible Explanation: The theoretical probability formula is \(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\).
## Section C: Word Problems Q15: A box contains 8 chocolate cookies and 12 vanilla cookies. If one cookie is selected at random, what is the probability that it is a chocolate cookie? Express your answer as a fraction in simplest form.
Solution:
Ans: Total number of cookies = 8 + 12 = 20 Number of chocolate cookies = 8 Probability of selecting a chocolate cookie = \(\frac{8}{20} = \frac{2}{5}\) Final Answer: \(\frac{2}{5}\)
Q16: Sarah is playing a game where she rolls a standard six-sided die. She wins if she rolls a 1 or a 6. What is the probability that Sarah wins on her next roll?
Solution:
Ans: Favorable outcomes (rolling 1 or 6) = 2 Total possible outcomes = 6 Probability of winning = \(\frac{2}{6} = \frac{1}{3}\) Final Answer: \(\frac{1}{3}\)
Q17: A spinner has 5 equal sections colored red, blue, green, yellow, and orange. If the spinner is spun once, what is the probability that it will NOT land on red?
Solution:
Ans: Total sections = 5 Sections that are NOT red = 4 (blue, green, yellow, orange) Probability of NOT landing on red = \(\frac{4}{5}\) Final Answer: \(\frac{4}{5}\)
Q18: In a class of 30 students, 18 are girls and 12 are boys. If one student is chosen at random to be class representative, what is the probability that a boy is chosen?
Solution:
Ans: Total number of students = 30 Number of boys = 12 Probability of choosing a boy = \(\frac{12}{30} = \frac{2}{5}\) Final Answer: \(\frac{2}{5}\)
Q19: A bag contains 15 balls: 6 are white, 5 are black, and 4 are yellow. If one ball is drawn at random, what is the probability that it is either white or yellow? Express your answer as a decimal.
Solution:
Ans: Total balls = 15 White or yellow balls = 6 + 4 = 10 Probability = \(\frac{10}{15} = \frac{2}{3}\) As a decimal: \(\frac{2}{3} \approx 0.67\) or 0.667 Final Answer: 0.67 (or 0.667)
Q20: Michael has a standard deck of 52 playing cards. He draws one card at random. What is the probability that the card is a heart? If the probability of this event NOT occurring is \(\frac{3}{4}\), verify this by calculating the probability directly.
Solution:
Ans: Total cards = 52 Number of hearts = 13 Probability of drawing a heart = \(\frac{13}{52} = \frac{1}{4}\) Probability of NOT drawing a heart = \(1 - \frac{1}{4} = \frac{3}{4}\) This confirms the given statement. Final Answer: \(\frac{1}{4}\) (probability of drawing a heart); \(\frac{3}{4}\) (probability of not drawing a heart) - verified
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