Experimental Probability exercise - Free Printable

Let's solve each problem step by step.

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Problem 1: Theoretical Probability of Rolling an Even Number

Question: What is the theoretical probability that an even number will be rolled on a number cube?

A standard number cube has six faces numbered 1 through 6. The even numbers are 2, 4, and 6. There are 3 even numbers out of 6 possible outcomes.

The theoretical probability is calculated as:
\[
P(\text{even number}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{6} = \frac{1}{2}
\]

Answer: \(\boxed{\frac{1}{2}}\)

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Problem 2: Experimental Probability of Rolling an Even Number

Question: What was the experimental probability of how many times an even number was actually rolled using the table?

From the table:
- Frequency of 2: 3
- Frequency of 4: 6
- Frequency of 6: 6

The total frequency of even numbers is:
\[
3 + 6 + 6 = 15
\]

The total number of rolls is the sum of all frequencies:
\[
8 + 3 + 9 + 6 + 4 + 6 = 36
\]

The experimental probability is:
\[
P(\text{even number}) = \frac{\text{Frequency of even numbers}}{\text{Total number of rolls}} = \frac{15}{36} = \frac{5}{12}
\]

Answer: \(\boxed{\frac{5}{12}}\)

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Problem 3: Expected Frequency of Rolling a 1 in 36 Rolls

Question: If you roll a number cube 36 times, how many times would you expect to roll the number one?

The theoretical probability of rolling a 1 is:
\[
P(1) = \frac{1}{6}
\]

If you roll the cube 36 times, the expected frequency is:
\[
\text{Expected frequency} = P(1) \times \text{Total rolls} = \frac{1}{6} \times 36 = 6
\]

Answer: \(\boxed{6}\)

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Problem 4: Actual Frequency of Rolling a 1

Question: How many times did you actually roll the number one in the experiment?

From the table, the frequency of rolling a 1 is:
\[
8
\]

Answer: \(\boxed{8}\)

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Problem 5: Theoretical Probability of Rolling a Number Greater Than 4

Question: What is the theoretical probability for rolling a number greater than 4?

The numbers greater than 4 are 5 and 6. There are 2 such numbers out of 6 possible outcomes.

The theoretical probability is:
\[
P(\text{number > 4}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{6} = \frac{1}{3}
\]

Answer: \(\boxed{\frac{1}{3}}\)

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Problem 6: Experimental Probability of Rolling a Number Greater Than 4

Question: What was the experimental probability of rolling a number greater than 4?

From the table:
- Frequency of 5: 4
- Frequency of 6: 6

The total frequency of numbers greater than 4 is:
\[
4 + 6 = 10
\]

The total number of rolls is 36 (as calculated earlier).

The experimental probability is:
\[
P(\text{number > 4}) = \frac{\text{Frequency of numbers > 4}}{\text{Total number of rolls}} = \frac{10}{36} = \frac{5}{18}
\]

Answer: \(\boxed{\frac{5}{18}}\)

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Problem 7: Difference Between Theoretical and Experimental Probability

Question: What is the difference between theoretical and experimental probability?

- Theoretical Probability: This is the expected probability based on mathematical calculations, assuming all outcomes are equally likely. It is determined before any experiment is conducted.
- Experimental Probability: This is the probability based on the results of an actual experiment or trial. It is calculated by dividing the number of times an event occurs by the total number of trials.

The key difference is that theoretical probability is a prediction, while experimental probability is based on observed data. Experimental probability may vary from theoretical probability due to randomness and the finite number of trials.

Answer: Theoretical probability is based on mathematical calculations and assumptions, while experimental probability is based on actual observations from experiments. Experimental probability can differ from theoretical probability due to randomness and the limited number of trials.

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Problem 8: Defective Cars

Question: If a car factory checks 360 cars and 8 of them have defects, how many will have defects out of 1260?

First, calculate the defect rate from the sample of 360 cars:
\[
\text{Defect rate} = \frac{\text{Number of defective cars}}{\text{Total number of cars checked}} = \frac{8}{360}
\]

Simplify the fraction:
\[
\frac{8}{360} = \frac{1}{45}
\]

Now, apply this defect rate to 1260 cars:
\[
\text{Expected number of defective cars} = \text{Defect rate} \times \text{Total number of cars} = \frac{1}{45} \times 1260 = 28
\]

Answer: \(\boxed{28}\)

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Final Answers:

1. \(\boxed{\frac{1}{2}}\)
2. \(\boxed{\frac{5}{12}}\)
3. \(\boxed{6}\)
4. \(\boxed{8}\)
5. \(\boxed{\frac{1}{3}}\)
6. \(\boxed{\frac{5}{18}}\)
7. Theoretical probability is based on mathematical calculations, while experimental probability is based on actual observations.
8. \(\boxed{28}\)