Q1: A fair coin is flipped 3 times. What is the probability of getting exactly 2 heads? (a) \(\frac{1}{8}\) (b) \(\frac{3}{8}\) (c) \(\frac{1}{2}\) (d) \(\frac{5}{8}\)
Solution:
Ans: (b) Explanation: The total number of outcomes when flipping a coin 3 times is \(2^3 = 8\). The favorable outcomes for exactly 2 heads are HHT, HTH, and THH, which gives us 3 outcomes. Therefore, the probability is \(\frac{3}{8}\).
Q2: In a simulation, you roll a six-sided die 60 times and record the frequency of each outcome. This type of simulation is used to estimate: (a) Theoretical probability (b) Experimental probability (c) Conditional probability (d) Independent events
Solution:
Ans: (b) Explanation:Experimental probability is based on actual trials or simulations. When you perform an experiment (rolling a die 60 times) and record outcomes, you are finding experimental probability. Theoretical probability is calculated without performing experiments.
Q3: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If you randomly select one marble, what is the probability of NOT selecting a blue marble? (a) \(\frac{3}{10}\) (b) \(\frac{7}{10}\) (c) \(\frac{1}{2}\) (d) \(\frac{2}{5}\)
Solution:
Ans: (b) Explanation: The total number of marbles is \(5 + 3 + 2 = 10\). The number of marbles that are NOT blue is \(5 + 2 = 7\). Therefore, the probability is \(\frac{7}{10}\). This is also equal to \(1 - \frac{3}{10} = \frac{7}{10}\).
Q4: Which of the following situations represents random outcomes? (a) The number of days in a week (b) The result of rolling a fair die (c) The boiling point of water at sea level (d) The number of sides on a triangle
Solution:
Ans: (b) Explanation: A random outcome is one where the result cannot be predicted with certainty before the event occurs. Rolling a fair die produces random outcomes because we cannot predict which number will appear. The other options are fixed values that do not involve randomness.
Q5: Two events A and B are independent if: (a) \(P(A \text{ and } B) = P(A) + P(B)\) (b) \(P(A \text{ and } B) = P(A) \times P(B)\) (c) \(P(A \text{ or } B) = P(A) \times P(B)\) (d) \(P(A) = P(B)\)
Solution:
Ans: (b) Explanation: Two events are independent if the occurrence of one does not affect the probability of the other. For independent events, \(P(A \text{ and } B) = P(A) \times P(B)\). Option (a) incorrectly adds probabilities, option (c) confuses "and" with "or," and option (d) is unrelated to independence.
Q6: A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability of spinning a number greater than 5? (a) \(\frac{1}{4}\) (b) \(\frac{3}{8}\) (c) \(\frac{1}{2}\) (d) \(\frac{5}{8}\)
Solution:
Ans: (b) Explanation: The numbers greater than 5 are 6, 7, and 8, which gives us 3 favorable outcomes. The total number of outcomes is 8. Therefore, the probability is \(\frac{3}{8}\).
Q7: In a probability simulation using random numbers, which of the following tools would be most appropriate to simulate flipping a coin 100 times? (a) A deck of cards (b) A random number generator producing 0s and 1s (c) A six-sided die (d) A spinner divided into three equal sections
Solution:
Ans: (b) Explanation: A coin has two equally likely outcomes (heads or tails). A random number generator that produces 0s and 1s can represent these two outcomes perfectly. A deck of cards, a six-sided die, and a three-section spinner do not have exactly two equally likely outcomes suitable for simulating a coin flip.
Q8: A student conducts a simulation by flipping a coin 50 times and gets 32 heads. The experimental probability of getting heads is: (a) \(\frac{32}{50}\) (b) \(\frac{1}{2}\) (c) \(\frac{18}{50}\) (d) \(\frac{32}{100}\)
Solution:
Ans: (a) Explanation:Experimental probability is calculated by dividing the number of favorable outcomes by the total number of trials. Here, heads occurred 32 times out of 50 flips, so the experimental probability is \(\frac{32}{50}\) or \(\frac{16}{25}\). Option (b) represents the theoretical probability, not the experimental result.
## Section B: Fill in the Blanks Q9: The set of all possible outcomes of a probability experiment is called the __________.
Solution:
Ans: sample space Explanation: The sample space is the complete set of all possible outcomes for a given probability experiment. For example, when rolling a die, the sample space is {1, 2, 3, 4, 5, 6}.
Q10: If an event is certain to happen, its probability is __________.
Solution:
Ans: 1 Explanation: A probability of 1 (or 100%) means the event will definitely occur. Probabilities range from 0 (impossible) to 1 (certain).
Q11: The probability of an event that cannot happen is __________.
Solution:
Ans: 0 Explanation: An impossible event has a probability of 0. For example, rolling a 7 on a standard six-sided die has probability 0.
Q12: A probability based on repeated trials of an experiment is called __________ probability.
Solution:
Ans: experimental Explanation:Experimental probability (or empirical probability) is determined by conducting actual experiments or simulations and recording the results.
Q13: When the outcome of one event does not affect the outcome of another event, the two events are said to be __________.
Solution:
Ans: independent Explanation:Independent events have no influence on each other. For example, flipping a coin and rolling a die are independent events.
Q14: The __________ probability is calculated using mathematical formulas and reasoning without performing experiments.
Solution:
Ans: theoretical Explanation:Theoretical probability is based on the expected outcomes using mathematical principles. For example, the theoretical probability of rolling a 3 on a fair die is \(\frac{1}{6}\).
## Section C: Word Problems Q15: A basketball player makes 7 out of 10 free throws during practice. Based on this experimental data, if she attempts 40 free throws in a game, how many would you expect her to make?
Solution:
Ans: Step 1: Find the experimental probability: \(\frac{7}{10} = 0.7\) Step 2: Multiply by the number of attempts: \(0.7 \times 40 = 28\) Final Answer: 28 free throws
Q16: A jar contains 12 chocolate chip cookies, 8 oatmeal cookies, and 10 sugar cookies. If you randomly select one cookie without looking, what is the probability of selecting a chocolate chip cookie? Express your answer as a simplified fraction.
Solution:
Ans: Step 1: Find total number of cookies: \(12 + 8 + 10 = 30\) Step 2: Calculate probability: \(\frac{12}{30} = \frac{2}{5}\) Final Answer: \(\frac{2}{5}\)
Q17: Maria conducts a simulation by rolling a standard six-sided die 120 times. She wants to determine the experimental probability of rolling a number less than 3. If she rolled a 1 twenty-five times and a 2 thirty times, what is the experimental probability of rolling a number less than 3?
Solution:
Ans: Step 1: Numbers less than 3 are 1 and 2 Step 2: Total favorable outcomes: \(25 + 30 = 55\) Step 3: Experimental probability: \(\frac{55}{120} = \frac{11}{24}\) Final Answer: \(\frac{11}{24}\)
Q18: A weather forecaster uses simulation to predict rain. In 200 simulations, rain occurred 85 times. Based on this simulation, what is the probability that it will rain tomorrow? Express your answer as a decimal rounded to two decimal places.
Solution:
Ans: Step 1: Calculate experimental probability: \(\frac{85}{200}\) Step 2: Convert to decimal: \(85 \div 200 = 0.425\) Step 3: Round to two decimal places: \(0.43\) Final Answer: 0.43 or 43%
Q19: A game spinner is divided into 5 equal sections colored red, blue, green, yellow, and orange. Josh spins the spinner twice. What is the probability that he gets red on the first spin AND blue on the second spin? Assume the spins are independent events.
Solution:
Ans: Step 1: Probability of red on first spin: \(\frac{1}{5}\) Step 2: Probability of blue on second spin: \(\frac{1}{5}\) Step 3: For independent events, multiply: \(\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}\) Final Answer: \(\frac{1}{25}\)
Q20: A cafeteria offers 3 types of sandwiches (turkey, ham, veggie) and 2 types of drinks (juice, water). If you randomly select one sandwich and one drink, what is the probability of getting a veggie sandwich and juice?
Solution:
Ans: Step 1: Total number of combinations: \(3 \times 2 = 6\) Step 2: Favorable outcome (veggie sandwich and juice): 1 combination Step 3: Probability: \(\frac{1}{6}\) Final Answer: \(\frac{1}{6}\)
The document Worksheet (with Solutions): Randomness, Probability, and Simulation is a part of the Grade 9 Course Statistics & Probability.
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