https://file.aiquickdraw.com/mathbot/file/1727495155564kciyyqys.jpg

Let's analyze and solve the given problem step by step based on the image you provided.

### Part (a): Theoretical Probability
The event in question is drawing either card 1, 3, or 4 and flipping heads. Here's how we calculate it:

- There are 4 cards, numbered 1, 2, 3, and 4. 
- The probability of drawing either a 1, 3, or 4 card is \( \frac{3}{4} \) because 3 of the 4 cards meet the condition.
- Since the coin is fair, the probability of flipping heads is \( \frac{1}{2} \).

Theoretical probability is calculated by multiplying the two independent probabilities:
\[
P(\text{1, 3, or 4 and heads}) = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} = 0.375
\]

Thus, the theoretical probability is **0.375**.

---

### Part (b): Experimental Probability
We now use the data from the table to calculate the experimental probability of this event.

- The relevant outcomes are 1H, 3H, and 4H.
- From the table:
  - Number of trials with outcome 1H = 134
  - Number of trials with outcome 3H = 133
  - Number of trials with outcome 4H = 134
- The total number of trials is 1050.

The experimental probability is:
\[
P(\text{1, 3, or 4 and heads}) = \frac{134 + 133 + 134}{1050} = \frac{401}{1050} \approx 0.382
\]

Thus, the experimental probability is **0.382**.

---

### Part (c): True Statement
Let's analyze the options:
- With a large number of trials, we expect the experimental probability to approach the theoretical probability, but they may not be exactly the same due to random variation. Therefore, **the second option is correct:**
  - "With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small."

---

Would you like further explanation on any part? Here are 5 related questions to expand on this:

1. How does increasing the number of trials affect the experimental probability?
2. How is theoretical probability different from experimental probability?
3. What are some examples of experiments where experimental probability might significantly differ from theoretical probability?
4. Can you calculate the probability of drawing a specific card (e.g., card 2) and flipping tails?
5. How can randomness affect experimental probability in small sample sizes?

**Tip:** When calculating experimental probabilities, always compare them with theoretical probabilities to assess the accuracy and reliability of your results.

There is a pack of four cards numbered 1 to 4. A coin is flipped, and the task is to find the theoretical and experimental probability of drawing certain cards and flipping heads, based on the outcomes of 1050 trials. The table shows the data from the trials.

Understand how to calculate both theoretical and experimental probabilities with a card draw and coin flip. This example analyzes outcomes from 1050 trials, providing detailed calculations for events like drawing cards numbered 1, 3, or 4 and flipping heads. Suitable for students in Grades 7-10.

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