Probability worksheet featuring coin flip and die roll experiments with theoretical and experimental probability questions.

Problem Analysis:

The worksheet contains two main sections: Coin Flip and Roll of the Die. Each section involves calculating probabilities, both theoretical and experimental, based on given data.

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Section A: Coin Flip

#### 1) What is the theoretical probability that the coin will land on tails?
- A fair coin has two equally likely outcomes: heads or tails.
- Theoretical probability is calculated as:
$$
P(\text{tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
$$
Here, there is 1 favorable outcome (tails) out of 2 possible outcomes (heads or tails).
$$
P(\text{tails}) = \frac{1}{2} = 0.5 \quad \text{(or 50%)}
$$

#### 2) What is the theoretical probability that the coin will land on heads?
- Similarly, for heads:
$$
P(\text{heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
$$
There is 1 favorable outcome (heads) out of 2 possible outcomes.
$$
P(\text{heads}) = \frac{1}{2} = 0.5 \quad \text{(or 50%)}
$$

#### 3) If the coin is flipped 140 times, how many times would you predict that the coin lands on heads?
- Using the theoretical probability of heads ($P(\text{heads}) = 0.5$), we predict the number of heads by multiplying the total number of flips by the probability:
$$
\text{Predicted number of heads} = 140 \times P(\text{heads}) = 140 \times 0.5 = 70
$$

#### 4) Johnny flipped a coin 450 times. His results are below:
| Heads | Tails |
|-------|-------|
| 340 | 210 |

- What is the experimental probability that the coin lands on heads?
- Experimental probability is calculated using the observed data:
$$
P_{\text{experimental}}(\text{heads}) = \frac{\text{Number of heads observed}}{\text{Total number of trials}}
$$
From the table, Johnny observed 340 heads out of 450 flips.
$$
P_{\text{experimental}}(\text{heads}) = \frac{340}{450} = \frac{34}{45} \approx 0.7556 \quad \text{(or about 75.56%)}
$$

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Section B: Roll of the Die

#### 5) \( P(4) = \)
- For a fair six-sided die, each face (1 through 6) has an equal probability of landing face up.
- Theoretical probability for any specific number (e.g., 4):
$$
P(4) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{6}
$$

#### 6) \( P(3 \text{ or } 5) = \)
- The event "3 or 5" includes two favorable outcomes: rolling a 3 or a 5.
- Theoretical probability:
$$
P(3 \text{ or } 5) = P(3) + P(5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
$$

#### 7) \( P(\text{not } 2) = \)
- The event "not 2" means any outcome except rolling a 2. There are 5 favorable outcomes (1, 3, 4, 5, 6).
- Theoretical probability:
$$
P(\text{not } 2) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{5}{6}
$$

#### 8) \( P(\text{odd}) = \)
- The odd numbers on a six-sided die are 1, 3, and 5. There are 3 favorable outcomes.
- Theoretical probability:
$$
P(\text{odd}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{3}{6} = \frac{1}{2}
$$

#### 9) If the die is rolled 300 times, how many times would you predict a roll of a 1 or a 6?
- The event "1 or 6" includes two favorable outcomes.
- Theoretical probability:
$$
P(1 \text{ or } 6) = P(1) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
$$
- Predicted number of times:
$$
\text{Predicted number} = 300 \times P(1 \text{ or } 6) = 300 \times \frac{1}{3} = 100
$$

#### 10) Johnny rolled the die 1,500 times. His results are below:
| Number on die | 1 | 2 | 3 | 4 | 5 | 6 |
|---------------|-----|-----|-----|-----|-----|-----|
| Times rolled | 230 | 245 | 300 | 280 | 215 | 230 |

- What is the experimental probability that the die lands on a 4?
- Experimental probability is calculated using the observed data:
$$
P_{\text{experimental}}(4) = \frac{\text{Number of times 4 was rolled}}{\text{Total number of rolls}}
$$
From the table, Johnny rolled a 4, 280 times out of 1,500 rolls.
$$
P_{\text{experimental}}(4) = \frac{280}{1500} = \frac{28}{150} = \frac{14}{75} \approx 0.1867 \quad \text{(or about 18.67%)}
$$

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Final Answers:

#### Section A: Coin Flip
1. \( P(\text{tails}) = \boxed{0.5} \)
2. \( P(\text{heads}) = \boxed{0.5} \)
3. Predicted number of heads in 140 flips: \( \boxed{70} \)
4. Experimental probability of heads: \( \boxed{0.7556} \)

#### Section B: Roll of the Die
5. \( P(4) = \boxed{\frac{1}{6}} \)
6. \( P(3 \text{ or } 5) = \boxed{\frac{1}{3}} \)
7. \( P(\text{not } 2) = \boxed{\frac{5}{6}} \)
8. \( P(\text{odd}) = \boxed{\frac{1}{2}} \)
9. Predicted number of times rolling a 1 or 6 in 300 rolls: \( \boxed{100} \)
10. Experimental probability of rolling a 4: \( \boxed{0.1867} \)

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Boxed Final Answer:

$$
\boxed{
\begin{aligned}
&\text{A1: } 0.5, \quad \text{A2: } 0.5, \quad \text{A3: } 70, \quad \text{A4: } 0.7556, \\
&\text{B5: } \frac{1}{6}, \quad \text{B6: } \frac{1}{3}, \quad \text{B7: } \frac{5}{6}, \quad \text{B8: } \frac{1}{2}, \\
&\text{B9: } 100, \quad \text{B10: } 0.1867.
\end{aligned}
}
$$