Students can use this worksheet to practice calculating probabilities based on frequency tables and spinner diagrams.

I'll work through each problem step by step, checking my calculations carefully.

Problem 1: What is the difference between theoretical and experimental probability?

Theoretical probability is what we calculate should happen based on math and logic (like knowing a coin has 2 sides, so heads = 1/2). Experimental probability is what actually happens when we do an experiment or trial many times.

Problems 2-5: Using the frequency table for rolling a die 20 times

Let me check the frequency table data:
- Roll 1: Frequency 3
- Roll 2: Frequency 4
- Roll 3: Frequency 3
- Roll 4: Frequency 5
- Roll 5: Frequency 3
- Roll 6: Frequency 2

Total rolls = 3 + 4 + 3 + 5 + 3 + 2 = 20 ✓

Problem 2: Find the theoretical probability of landing on 3.

A fair die has 6 faces numbered 1-6. Each face has equal chance.
Theoretical probability = favorable outcomes / total possible outcomes
P(rolling a 3) = 1/6

Problem 3: Find the experimental probability of landing on 3.

From the table: Number of times 3 appeared = 3
Total rolls = 20
Experimental probability = 3/20

Problem 4: Find the theoretical probability of landing on either 5 or 6.

Favorable outcomes: rolling a 5 OR rolling a 6 = 2 outcomes
Total possible outcomes = 6
P(5 or 6) = 2/6 = 1/3

Problem 5: Find the experimental probability of landing on either 5 or 6.

From the table:
- Number of times 5 appeared = 3
- Number of times 6 appeared = 2
- Total times 5 or 6 appeared = 3 + 2 = 5
Total rolls = 20
Experimental probability = 5/20 = 1/4

Problems 6-9: Spinner problems

Looking at the spinner, it's divided into 8 equal sections labeled A through H.

Problem 6: What is the theoretical probability of landing on C?

Total sections = 8
Sections with C = 1
Theoretical probability = 1/8

Problem 7: What is the experimental probability of landing on C?

Given: Spun 30 times, landed on C 8 times
Experimental probability = 8/30 = 4/15

Problem 8: What would you expect to happen to the experimental probability as you spin more and more times?

As you increase the number of trials, the experimental probability gets closer to the theoretical probability. This is called the Law of Large Numbers.

Problem 9: If you were to spin the spinner 1,000 times, how many times would you expect it to land on C? Would you expect the fraction to be closer to 1/8 than 8/30?

Expected number of times landing on C = (theoretical probability) × (number of spins)
= (1/8) × 1000 = 125 times

Yes, with 1,000 spins, the experimental probability would be much closer to the theoretical probability of 1/8 than the result from only 30 spins (8/30). More trials give results closer to theoretical probability.

Problem 10: Drawing marbles from a bag

Bag contains: 3 red, 2 blue, 1 green, 1 yellow = 7 marbles total

a) P(red) = 3/7

b) P(blue) = 2/7

c) P(green) = 1/7

d) P(yellow) = 1/7

e) P(not yellow) = 1 - P(yellow) = 1 - 1/7 = 6/7
Or: (red + blue + green) / total = (3 + 2 + 1) / 7 = 6/7

f) P(red or blue) = P(red) + P(blue) = 3/7 + 2/7 = 5/7

g) P(not green) = 1 - P(green) = 1 - 1/7 = 6/7
Or: (red + blue + yellow) / total = (3 + 2 + 1) / 7 = 6/7

Final Answer:

1. Theoretical probability is calculated using math; experimental probability comes from actual trials.

2. 1/6

3. 3/20

4. 1/3

5. 1/4

6. 1/8

7. 4/15

8. It would get closer to the theoretical probability.

9. 125 times; Yes, it would be closer to 1/8.

10. a) 3/7 b) 2/7 c) 1/7 d) 1/7 e) 6/7 f) 5/7 g) 6/7