Question 1: A survey shows 30 students like apples, 20 like oranges, and 10 like both. Fill in the following table: Number of students liking Apples only, Oranges only, both, and none.

Answer: {"both":10,"none":60,"apples_only":20,"oranges_only":10}

Question 2: Given the table of students' preferences, what is the probability that a randomly selected student likes apples but not oranges?

Answer: 20/100

Question 3: Construct a probability table for two independent events, where event A has probability 0.4 and event B has probability 0.6. Fill in all four joint probabilities.

Answer: {"A_and_B":0.24,"A_and_not_B":0.16,"not_A_and_B":0.36,"not_A_and_not_B":0.24}

Question 4: In a box of 50 chocolates, 20 are caramel, 15 are hazelnut, and 10 are both caramel and hazelnut. Fill the table with the counts for each category.

Answer: {"both":10,"none":25,"caramel_only":10,"hazelnut_only":5}

Question 5: Using the chocolates table, what is the probability that a randomly selected chocolate is not caramel?

Answer: 25/50

Question 6: A survey shows 40% of students prefer tea, 35% prefer coffee, and 10% prefer both. Draw a probability table representing these preferences and their joint probabilities.

Answer: {"both":0.1,"neither":0.35,"tea_only":0.3,"coffee_only":0.25}

Question 7: In the tea and coffee preferences table, what is the probability that a student prefers tea only?

Answer: 0.30

Question 8: A bag contains 8 red, 6 blue, and 4 green marbles. If two marbles are drawn at random without replacement, complete the table for possible outcomes: both red, both blue, both green, and mixed colors.

Answer: {"red_red":28,"red_blue":48,"blue_blue":15,"red_green":32,"blue_green":24,"green_green":3}

Question 9: Using the marble outcome counts, what is the probability of drawing one red and one blue marble in any order?

Answer: 48/91

Question 10: A spinner has 3 red, 4 yellow, and 5 green sections. Construct a probability table for selecting each color and their joint probabilities for two spins.

Answer: {"red_red":0.09,"red_green":0.15,"red_yellow":0.12,"green_green":0.25,"yellow_green":0.2,"yellow_yellow":0.16}

Question 11: Identify the common mistake in constructing joint probability tables and explain how to correct it.

Answer: Students often forget to ensure that the sum of all joint probabilities equals 1. To correct this, verify the total sum and adjust values proportionally if necessary.

Question 12: Given a probability table where the total sums to 0.95 instead of 1, what is the most likely mistake made, and how can it be fixed?

Answer: The total probability sum is less than 1, indicating missing or unaccounted outcomes. To fix this, ensure all outcomes are included and probabilities are correctly assigned so that the total sums to 1.