Math 7 worksheet focusing on theoretical and experimental probability, including real-world applications like survey data and spinner outcomes.

Here are the step-by-step solutions for the problems on the worksheet.

Problem 1: Number Cube

a. Find the experimental probability of landing on a 3.
* Step 1: Identify the total number of trials (rolls). The problem states the cube was rolled 24 times.
* Step 2: Identify the number of successful outcomes (landing on a 3). The table shows this happened 6 times.
* Step 3: Write the probability as a fraction: $\frac{\text{successful outcomes}}{\text{total trials}} = \frac{6}{24}$.
* Step 4: Simplify the fraction. Both numbers can be divided by 6.
* $6 \div 6 = 1$
* $24 \div 6 = 4$
* Result: $\frac{1}{4}$

b. Find the experimental probability of not landing on a 6.
* Step 1: Identify how many times it landed on a 6. The table shows 8 times.
* Step 2: Calculate how many times it did *not* land on a 6. Subtract the 6s from the total rolls: $24 - 8 = 16$.
* Step 3: Write the probability: $\frac{16}{24}$.
* Step 4: Simplify the fraction. Both numbers can be divided by 8.
* $16 \div 8 = 2$
* $24 \div 8 = 3$
* Result: $\frac{2}{3}$

c. Compare the experimental probability... to its theoretical probability.
* Step 1: Find the theoretical probability of rolling a 3. A standard cube has 6 sides, and only one side is a 3. So, the theoretical probability is $\frac{1}{6}$.
* Step 2: Compare it to the experimental probability from part (a), which is $\frac{1}{4}$.
* Step 3: To compare, find a common denominator (12 works well).
* Experimental: $\frac{1}{4} = \frac{3}{12}$
* Theoretical: $\frac{1}{6} = \frac{2}{12}$
* Conclusion: Since $\frac{3}{12}$ is larger than $\frac{2}{12}$, the experimental probability is higher than the theoretical probability.

---

Problem 2: Spinner Experiment

a. What is the probability that the spinner will stop on blue?
* Step 1: Find the total number of spins by adding up all the frequencies in the table:
* $40 \text{ (Blue)} + 25 \text{ (Green)} + 10 \text{ (Red)} + 5 \text{ (Yellow)} = 80 \text{ total spins}$.
* Step 2: Find the frequency for Blue, which is 40.
* Step 3: Write the fraction: $\frac{40}{80}$.
* Step 4: Simplify. 40 is exactly half of 80.
* Result: $\frac{1}{2}$

b. Out of 500 people surveyed, how many would you expect combined reading books or surfing the Internet at their convenience level?
* Step 1: Find the percent for "Reading Books" (25%) and "Surfing the Internet" (5%).
* Step 2: Add these percentages together: $25\% + 5\% = 30\%$.
* Step 3: Convert the percentage to a decimal: $30\% = 0.30$.
* Step 4: Multiply by the total number of people surveyed (500):
* $500 \times 0.30 = 150$.
* Result: 150 people

c. Out of 700 people surveyed, is it reasonable to expect that 140 combined reading newspapers and watching television? Why or why not?
* Step 1: Find the percent for "Reading Newspapers" (10%) and "Watching Television" (5%).
* Step 2: Add them together: $10\% + 5\% = 15\%$.
* Step 3: Calculate the expected number out of 700 people:
* $15\%$ of $700 = 0.15 \times 700$.
* $10\%$ of $700$ is $70$. Half of that ($5\%$) is $35$.
* $70 + 35 = 105$.
* Step 4: Compare the calculated expectation (105) with the number in the question (140).
* Conclusion: No, it is not reasonable. You would expect about 105 people, so 140 is much higher than expected.

---

Problem 3: Game Marker

a. Find the experimental probability of landing on green.
* Step 1: Find the total number of spins by adding the frequencies:
* $10 \text{ (Green)} + 15 \text{ (Blue)} + 25 \text{ (Yellow)} + 60 \text{ (Red)} = 110 \text{ total spins}$.
* Step 2: Find the frequency for Green, which is 10.
* Step 3: Write the fraction: $\frac{10}{110}$.
* Step 4: Simplify by dividing top and bottom by 10.
* Result: $\frac{1}{11}$

b. Find the experimental probability of landing on red.
* Step 1: Use the total spins calculated above: 110.
* Step 2: Find the frequency for Red, which is 60.
* Step 3: Write the fraction: $\frac{60}{110}$.
* Step 4: Simplify by dividing top and bottom by 10.
* Result: $\frac{6}{11}$

c. If the spinner is spun 50 more times, how many of those spins would you expect the pointer to land on blue?
* Step 1: Find the experimental probability of landing on Blue using the original data.
* Frequency of Blue = 15. Total spins = 110.
* Probability = $\frac{15}{110}$.
* Step 2: Multiply this probability by the new number of spins (50).
* $\frac{15}{110} \times 50$
* First, simplify $\frac{15}{110}$ to $\frac{3}{22}$.
* $\frac{3}{22} \times 50 = \frac{150}{22}$.
* $150 \div 22 \approx 6.81$.
* Step 3: Round to the nearest whole number. Since you can't have a fraction of a spin result, we round 6.81 to 7.
* Result: 7 spins

Final Answer:
1a. 1/4
1b. 2/3
1c. The experimental probability (1/4) is higher than the theoretical probability (1/6).

2a. 1/2
2b. 150 people
2c. No, it is not reasonable. Based on the percentages (15%), you would expect 105 people, not 140.

3a. 1/11
3b. 6/11
3c. 7 spins