Question 1: A bag contains 3 red and 2 blue balls. Using a grid, represent all possible outcomes when randomly selecting a ball twice (with replacement).

Answer: Red-Red, Red-Blue, Blue-Red, Blue-Blue

Question 2: In a grid, shade the cells representing outcomes where the first event is a success and the second is a failure, given that two events are independent with probabilities p=0.6 and q=0.4 respectively.

Answer: Outcome cells: success-failure

Question 3: Construct a 3x3 grid to show all outcomes when flipping a coin twice, coding Heads as H and Tails as T.

Answer: HH, HT, TH, TT

Question 4: Using the grid, find the probability that exactly one of the two events occurs, given the outcomes shown.

Answer: Outcomes: success-failure, failure-success; probability = 2/9

Question 5: A survey shows that 30% of students like apples and 50% like oranges. Draw a grid to represent the joint probabilities if 70% like neither, assuming independence.

Answer: Constructed grid with probabilities: P(neither)=0.7, P(apple only)=0.3*0.5=0.15, P(orange only)=0.5*0.7=0.35, P(both)=0.3*0.5=0.15

Question 6: A grid shows the probabilities of drawing a red or black card and winning or losing a game. How do you find the probability of drawing a red card and winning?

Answer: Multiply the probability of red and winning cells

Question 7: Explain why the sum of all probabilities in a grid for two events must be 1, and identify any common mistake when shading cells.

Answer: Total probability must be 1; common mistake: double-counting overlapping outcomes

Question 8: Given a grid with specific joint probabilities, determine if the events are independent using the product rule. Show your working.

Answer: Compare P(A and B) with P(A)*P(B); if equal, independent

Question 9: A real-world scenario: A school surveys students about liking sports (S) and music (M). Draw a grid representing the data, where 40% like sports, 30% like music, and 10% like both.

Answer: Construct grid with probabilities: P(both)=0.10, P(only S)=0.30-0.10=0.20, P(only M)=0.30-0.10=0.20, P(neither)=1-0.40-0.20-0.20=0.20

Question 10: A student creates a grid with outcomes for two events, but the total probability sums to 1.2. What mistake might they have made, and how can it be corrected?

Answer: Mistake: double-counting or incorrect probabilities; correction: ensure sum equals 1 by adjusting probabilities

Question 11: Challenge: Construct a 4x4 grid for two independent events, each with 4 equally likely outcomes. Calculate the probability that both events produce specific outcomes, and explain your reasoning.

Answer: Each outcome probability: 1/4; joint probability: (1/4)*(1/4)=1/16