Math Problem Statement

Use the given experimental data and theoretical concepts to calculate the probabilities for both drawing the 4 card and flipping tails, then choose the correct statement about experimental and theoretical probabilities.

Solution

Let's analyze the data and solve the questions based on the given information:


(a) Experimental Probability

We are asked to calculate the experimental probability of the event "drawing the 4 card and flipping tails." The outcomes relevant to this event are labeled 4T, and the number of trials for this event is 6.

The total number of trials across all outcomes is: 10+11+8+9+7+13+6+6=7010 + 11 + 8 + 9 + 7 + 13 + 6 + 6 = 70

The experimental probability P(4 and T)P(\text{4 and T}) is given by: P(4 and T)=Number of trials for 4TTotal number of trials=670=0.086P(\text{4 and T}) = \frac{\text{Number of trials for 4T}}{\text{Total number of trials}} = \frac{6}{70} = 0.086


(b) Theoretical Probability

Here, we assume that the card is chosen randomly and the coin is fair. There are 4 cards and 2 sides on the coin (heads or tails), making a total of 4×2=84 \times 2 = 8 possible outcomes, all equally likely.

The theoretical probability of drawing the 4 card and flipping tails is: P(4 and T)=18=0.125P(\text{4 and T}) = \frac{1}{8} = 0.125


(c) True Statement

The correct answer is: "As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal."\text{"As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal."} This is based on the law of large numbers, which states that experimental probabilities tend to approach theoretical probabilities as the number of trials increases.


Final Answers

  1. (a) Experimental Probability: 0.086\mathbf{0.086}
  2. (b) Theoretical Probability: 0.125\mathbf{0.125}
  3. (c) Correct Statement: "As the number of trials increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal."

Would you like me to explain any part further, or do calculations in more detail?

Relative Questions:

  1. How is experimental probability different from theoretical probability?
  2. Why do we divide by the total number of trials when calculating probability?
  3. Can the experimental probability ever equal the theoretical probability exactly?
  4. What happens if the number of trials is very small? How reliable is the experimental probability?
  5. What is the probability of drawing a specific card (e.g., 2 or 3) and flipping heads?

Tip:

When solving probability problems, always verify whether outcomes are equally likely. If they aren't, the theoretical probability will need adjustment!

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Math Problem Analysis

Mathematical Concepts

Probability
Experimental Probability
Theoretical Probability

Formulas

Experimental Probability = (Favorable Outcomes) / (Total Outcomes)
Theoretical Probability = (Number of Favorable Outcomes) / (Total Possible Outcomes)

Theorems

Law of Large Numbers

Suitable Grade Level

Grades 9-11