Question 1: Calculate the probability of rolling a number greater than 4 on a fair six-sided die.

Answer: 1/3

Question 2: A die is rolled twice. What is the probability that both rolls show the same number?

Answer: 1/6

Question 3: A student claims that rolling a 3 on a die is more likely than rolling a 6. Is this true? Explain your reasoning.

Answer: No, both outcomes have equal probability of 1/6.

Question 4: A die is biased such that the probability of rolling a 4 is 1/2. Is this a fair die? Explain why or why not.

Answer: No, because a fair die has equal probabilities for all faces, each 1/6.

Question 5: If a die is rolled three times, what is the probability that the sum of the three rolls is exactly 10?

Answer: 27/216 or 1/8

Question 6: A student calculates the probability of rolling an even number on a die as 1/2. Is this correct? Identify any mistake and correct it if necessary.

Answer: Yes, since even numbers are 2, 4, 6; probability is 3/6=1/2. No mistake.

Question 7: Construct a probability tree diagram for two rolls of a die, illustrating all possible outcomes and their probabilities.

Answer: Student draws a tree with 36 outcomes, each with probability 1/36.

Question 8: A bag contains two dice: one fair and one biased (favoring 6). If a die is chosen at random and rolled, what is the probability of rolling a 6?

Answer: Probability = (1/2)*(1/6) + (1/2)*(5/12) = 1/12 + 5/24 = 7/24

Question 9: Identify and correct the mistake in this student’s reasoning: 'Since there are 6 outcomes on a die, the probability of rolling any specific number is 1/6, so the probability of rolling an even number is 3/6 = 1/2.'

Answer: No mistake; the calculation is correct.

Question 10: A die is rolled four times. Calculate the probability that all four rolls are different numbers.

Answer: P = (6/6)*(5/6)*(4/6)*(3/6) = 360/1296 = 5/18

Question 11: A player rolls two dice 100 times. For each roll, they record if at least one die shows a 5. What is the expected number of such outcomes?

Answer: Expected = 100 * P(at least one 5) = 100 * (1 - (5/6)*(5/6)) = 100 * (1 - 25/36) = 100 * 11/36 ≈ 30.56