Question
Flipped 3 pennies and did this 100 times. You got 3 tails 40 times. What is the difference between the experimental probability of getting 3 tails and the theoretical probability? (simplified fraction.)
Answers
GPT-4o mini
To find the difference between the experimental probability and the theoretical probability of getting 3 tails when flipping 3 pennies, we first need to calculate both probabilities.
### 1. Experimental Probability
The experimental probability of an event is calculated using the formula:
\[
P(E) = \frac{\text{Number of times the event occurred}}{\text{Total trials}}
\]
In your case, you flipped 3 pennies 100 times and got 3 tails 40 times. Thus, the experimental probability \( P(E) \) is:
\[
P(E) = \frac{40}{100} = 0.4
\]
### 2. Theoretical Probability
The theoretical probability of getting 3 tails when flipping 3 pennies can be calculated as follows. Each penny has 2 possible outcomes: heads (H) or tails (T). When flipping 3 pennies, the total number of outcomes is:
\[
2^3 = 8
\]
The only outcome that results in getting 3 tails is TTT. Thus, there is 1 favorable outcome (getting 3 tails). Therefore, the theoretical probability \( P(T) \) is:
\[
P(T) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{8}
\]
### 3. Difference Between Probabilities
Now, we need to find the difference between the experimental probability and the theoretical probability. We have:
- Experimental probability: \( P(E) = \frac{40}{100} = \frac{4}{10} = \frac{2}{5} \)
- Theoretical probability: \( P(T) = \frac{1}{8} \)
To find the difference \( P(E) - P(T) \), we first convert both fractions to a common denominator, which is 40.
The experimental probability becomes:
\[
P(E) = \frac{2}{5} = \frac{16}{40}
\]
The theoretical probability converts to:
\[
P(T) = \frac{1}{8} = \frac{5}{40}
\]
Now we can compute the difference:
\[
P(E) - P(T) = \frac{16}{40} - \frac{5}{40} = \frac{16 - 5}{40} = \frac{11}{40}
\]
### Conclusion
The difference between the experimental probability of getting 3 tails and the theoretical probability is:
\[
\frac{11}{40}
\]
This fraction is already in its simplest form.
### 1. Experimental Probability
The experimental probability of an event is calculated using the formula:
\[
P(E) = \frac{\text{Number of times the event occurred}}{\text{Total trials}}
\]
In your case, you flipped 3 pennies 100 times and got 3 tails 40 times. Thus, the experimental probability \( P(E) \) is:
\[
P(E) = \frac{40}{100} = 0.4
\]
### 2. Theoretical Probability
The theoretical probability of getting 3 tails when flipping 3 pennies can be calculated as follows. Each penny has 2 possible outcomes: heads (H) or tails (T). When flipping 3 pennies, the total number of outcomes is:
\[
2^3 = 8
\]
The only outcome that results in getting 3 tails is TTT. Thus, there is 1 favorable outcome (getting 3 tails). Therefore, the theoretical probability \( P(T) \) is:
\[
P(T) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{8}
\]
### 3. Difference Between Probabilities
Now, we need to find the difference between the experimental probability and the theoretical probability. We have:
- Experimental probability: \( P(E) = \frac{40}{100} = \frac{4}{10} = \frac{2}{5} \)
- Theoretical probability: \( P(T) = \frac{1}{8} \)
To find the difference \( P(E) - P(T) \), we first convert both fractions to a common denominator, which is 40.
The experimental probability becomes:
\[
P(E) = \frac{2}{5} = \frac{16}{40}
\]
The theoretical probability converts to:
\[
P(T) = \frac{1}{8} = \frac{5}{40}
\]
Now we can compute the difference:
\[
P(E) - P(T) = \frac{16}{40} - \frac{5}{40} = \frac{16 - 5}{40} = \frac{11}{40}
\]
### Conclusion
The difference between the experimental probability of getting 3 tails and the theoretical probability is:
\[
\frac{11}{40}
\]
This fraction is already in its simplest form.