Theoretical Vs Experimental Probability 2010-2024 Form - Fill Out ... - Free Printable

Problem Analysis:

The problem involves calculating both experimental probability (based on observed data) and theoretical probability (based on known distributions). Let's break it down step by step.

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#### Part 1: Experimental Probability of Selecting a Heart
From the table provided in the image, we can see the recorded results of Amanda's experiment:

- Diamonds: 8
- Hearts: 14
- Spades: 16
- Clubs: 12

The total number of trials is:
$$
8 + 14 + 16 + 12 = 50
$$

The experimental probability of selecting a heart is calculated as:
$$
P(\text{Heart}) = \frac{\text{Number of Hearts}}{\text{Total Trials}} = \frac{14}{50}
$$

Simplify the fraction:
$$
P(\text{Heart}) = \frac{7}{25}
$$

#### Solution for Part 1:
$$
\boxed{\frac{7}{25}}
$$

---

#### Part 2: Theoretical Probability of Selecting a Heart
A standard deck of 52 cards has 13 hearts. The theoretical probability of selecting a heart is:
$$
P(\text{Heart}) = \frac{\text{Number of Hearts in Deck}}{\text{Total Cards in Deck}} = \frac{13}{52}
$$

Simplify the fraction:
$$
P(\text{Heart}) = \frac{1}{4}
$$

#### Solution for Part 2:
$$
\boxed{\frac{1}{4}}
$$

---

#### Part 3: Experimental Probability of Selecting a Diamond or a Spade
From the table:
- Diamonds: 8
- Spades: 16

The total number of diamonds or spades is:
$$
8 + 16 = 24
$$

The experimental probability of selecting a diamond or a spade is:
$$
P(\text{Diamond or Spade}) = \frac{\text{Number of Diamonds or Spades}}{\text{Total Trials}} = \frac{24}{50}
$$

Simplify the fraction:
$$
P(\text{Diamond or Spade}) = \frac{12}{25}
$$

#### Solution for Part 3:
$$
\boxed{\frac{12}{25}}
$$

---

#### Part 4: Theoretical Probability of Selecting a Diamond or a Spade
A standard deck of 52 cards has 13 diamonds and 13 spades. The theoretical probability of selecting a diamond or a spade is:
$$
P(\text{Diamond or Spade}) = \frac{\text{Number of Diamonds} + \text{Number of Spades}}{\text{Total Cards in Deck}} = \frac{13 + 13}{52}
$$

Simplify the fraction:
$$
P(\text{Diamond or Spade}) = \frac{26}{52} = \frac{1}{2}
$$

#### Solution for Part 4:
$$
\boxed{\frac{1}{2}}
$$

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#### Part 5: Compare Results and Describe Findings
- Experimental Probability of Heart: $\frac{7}{25} = 0.28$
- Theoretical Probability of Heart: $\frac{1}{4} = 0.25$

- Experimental Probability of Diamond or Spade: $\frac{12}{25} = 0.48$
- Theoretical Probability of Diamond or Spade: $\frac{1}{2} = 0.50$

Comparison:
- The experimental probability of selecting a heart ($0.28$) is slightly higher than the theoretical probability ($0.25$).
- The experimental probability of selecting a diamond or spade ($0.48$) is slightly lower than the theoretical probability ($0.50$).

Findings:
- The experimental probabilities are close to the theoretical probabilities but show slight variations due to the randomness inherent in the experiment. With more trials, the experimental probabilities would likely converge closer to the theoretical values.

#### Solution for Part 5:
$$
\boxed{\text{The experimental probabilities are close to the theoretical probabilities but show minor deviations due to randomness.}}
$$

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#### Part 6: Eye Color Survey
The survey results for eye color distribution in Dale's class are given in the table:

| Eye Color | Blue | Brown | Green | Hazel |
|-----------|------|-------|-------|-------|
| Number | 12 | 15 | 4 | 9 |

The total number of students surveyed is:
$$
12 + 15 + 4 + 9 = 40
$$

##### a. Experimental Probability Distribution for Each Eye Color
- Blue:
$$
P(\text{Blue}) = \frac{\text{Number of Blue Eyes}}{\text{Total Students}} = \frac{12}{40} = \frac{3}{10}
$$

- Brown:
$$
P(\text{Brown}) = \frac{\text{Number of Brown Eyes}}{\text{Total Students}} = \frac{15}{40} = \frac{3}{8}
$$

- Green:
$$
P(\text{Green}) = \frac{\text{Number of Green Eyes}}{\text{Total Students}} = \frac{4}{40} = \frac{1}{10}
$$

- Hazel:
$$
P(\text{Hazel}) = \frac{\text{Number of Hazel Eyes}}{\text{Total Students}} = \frac{9}{40}
$$

##### b. Experimental Probability of a Student Having Blue or Green Eyes
The number of students with blue or green eyes is:
$$
12 + 4 = 16
$$

The experimental probability is:
$$
P(\text{Blue or Green}) = \frac{\text{Number of Blue or Green Eyes}}{\text{Total Students}} = \frac{16}{40} = \frac{2}{5}
$$

##### c. Experimental Probability of a Student Not Having Green or Hazel Eyes
The number of students with green or hazel eyes is:
$$
4 + 9 = 13
$$

The number of students not having green or hazel eyes is:
$$
40 - 13 = 27
$$

The experimental probability is:
$$
P(\text{Not Green or Hazel}) = \frac{\text{Number of Students Not Green or Hazel}}{\text{Total Students}} = \frac{27}{40}
$$

##### d. Expected Number of Brown-Eyed Students in Dale's Grade
If the distribution of eye colors in Dale's class is similar to the distribution in his grade, and there are 360 students in his grade, the expected number of brown-eyed students is:
$$
\text{Expected Number} = P(\text{Brown}) \times \text{Total Students in Grade}
$$

From part (a), $P(\text{Brown}) = \frac{15}{40} = \frac{3}{8}$. Therefore:
$$
\text{Expected Number} = \frac{3}{8} \times 360 = 135
$$

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#### Final Answers for Part 6:
##### a.
$$
P(\text{Blue}) = \frac{3}{10}, \quad P(\text{Brown}) = \frac{3}{8}, \quad P(\text{Green}) = \frac{1}{10}, \quad P(\text{Hazel}) = \frac{9}{40}
$$

##### b.
$$
P(\text{Blue or Green}) = \frac{2}{5}
$$

##### c.
$$
P(\text{Not Green or Hazel}) = \frac{27}{40}
$$

##### d.
$$
\boxed{135}
$$

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Final Summary of All Solutions:

1. $\boxed{\frac{7}{25}}$
2. $\boxed{\frac{1}{4}}$
3. $\boxed{\frac{12}{25}}$
4. $\boxed{\frac{1}{2}}$
5. $\boxed{\text{The experimental probabilities are close to the theoretical probabilities but show minor deviations due to randomness.}}$
6.
- a. $P(\text{Blue}) = \frac{3}{10}$, $P(\text{Brown}) = \frac{3}{8}$, $P(\text{Green}) = \frac{1}{10}$, $P(\text{Hazel}) = \frac{9}{40}$
- b. $\boxed{\frac{2}{5}}$
- c. $\boxed{\frac{27}{40}}$
- d. $\boxed{135}$