Probability Worksheet 4 - Experimental vs. Theoretical - Ellie ... - Free Printable

Let's solve each problem step by step, starting with the card experiment and then moving to Dale’s eye color survey.

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Part 1: Amanda’s Card Experiment

We are given a tally chart of suits selected from a standard 52-card deck. Each time a card is drawn, it's replaced, so the trials are independent.

First, let's count how many times each suit was picked based on the tally marks:

- Diamonds: |||| ||| → 5 + 3 = 8
- Hearts: |||| ||| ||| → 5 + 3 + 3 = 11
- Spades: |||| ||| |||| | → 5 + 3 + 5 + 1 = 14
- Clubs: ||| → 3

Total number of trials = 8 + 11 + 14 + 3 = 36

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#### 1. Experimental probability of selecting a heart

Experimental probability = (Number of times hearts occurred) / (Total trials)

$$
P(\text{Heart}) = \frac{11}{36}
$$

✔ Answer: $ \frac{11}{36} $

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#### 2. Theoretical probability of selecting a heart

In a standard deck, there are 13 hearts out of 52 cards.

$$
P(\text{Heart}) = \frac{13}{52} = \frac{1}{4}
$$

✔ Answer: $ \frac{1}{4} $ or 0.25

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#### 3. Experimental probability of selecting a diamond or a spade

Add the number of diamonds and spades:
- Diamonds: 8
- Spades: 14
- Total favorable outcomes: 8 + 14 = 22

$$
P(\text{Diamond or Spade}) = \frac{22}{36} = \frac{11}{18}
$$

✔ Answer: $ \frac{11}{18} $

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#### 4. Theoretical probability of selecting a diamond or a spade

There are 13 diamonds and 13 spades in a deck → 26 total.

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

✔ Answer: $ \frac{1}{2} $ or 0.5

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#### 5. Compare these results and describe your findings

- Experimental:
- Heart: $ \frac{11}{36} \approx 0.3056 $
- Diamond or Spade: $ \frac{11}{18} \approx 0.6111 $

- Theoretical:
- Heart: $ \frac{1}{4} = 0.25 $
- Diamond or Spade: $ \frac{1}{2} = 0.5 $

Observations:
- The experimental probabilities are somewhat close to theoretical ones, but not exact.
- Hearts were selected more often than expected (30.56% vs 25%), while black suits (spades and clubs) were selected more frequently overall.
- The experimental probability of getting a diamond or spade is higher than theoretical (61.1% vs 50%), likely due to random variation in small sample size (only 36 trials).

👉 Conclusion: With only 36 trials, experimental results may differ from theoretical due to chance. As the number of trials increases, we expect the experimental probabilities to approach the theoretical values.

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Part 2: Dale’s Eye Color Survey

Given data:

| Eye Color | Blue | Brown | Green | Hazel |
|-----------|------|-------|--------|--------|
| Number | 12 | 58 | 2 | 8 |

Total students surveyed = 12 + 58 + 2 + 8 = 80

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#### 6a. Find the experimental probability distribution for each eye color

$$
P(\text{Blue}) = \frac{12}{80} = \frac{3}{20} = 0.15
$$

$$
P(\text{Brown}) = \frac{58}{80} = \frac{29}{40} = 0.725
$$

$$
P(\text{Green}) = \frac{2}{80} = \frac{1}{40} = 0.025
$$

$$
P(\text{Hazel}) = \frac{8}{80} = \frac{1}{10} = 0.1
$$

✔ Answers:
- P(blue) = $ \frac{3}{20} $ or 0.15
- P(brown) = $ \frac{29}{40} $ or 0.725
- P(green) = $ \frac{1}{40} $ or 0.025
- P(hazel) = $ \frac{1}{10} $ or 0.1

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#### 6b. Probability that a student has blue or green eyes

Add the probabilities:

$$
P(\text{Blue or Green}) = P(\text{Blue}) + P(\text{Green}) = \frac{12}{80} + \frac{2}{80} = \frac{14}{80} = \frac{7}{40} = 0.175
$$

✔ Answer: $ \frac{7}{40} $ or 0.175

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#### 6c. Probability that a student does NOT have green or hazel eyes

This means the student has blue or brown eyes.

So:

$$
P(\text{Not Green or Hazel}) = 1 - P(\text{Green or Hazel})
$$

$$
P(\text{Green or Hazel}) = \frac{2}{80} + \frac{8}{80} = \frac{10}{80} = \frac{1}{8}
$$

$$
P(\text{Not Green or Hazel}) = 1 - \frac{1}{8} = \frac{7}{8}
$$

Alternatively, add blue and brown:

$$
\frac{12 + 58}{80} = \frac{70}{80} = \frac{7}{8}
$$

✔ Answer: $ \frac{7}{8} $ or 0.875

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#### 6d. How many of 360 students would be expected to have brown eyes?

From the survey, P(brown) = $ \frac{58}{80} = 0.725 $

Expected number in 360 students:

$$
0.725 \times 360 = 261
$$

✔ Answer: 261 students

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✔ Final Answers Summary:

#### Card Problems:

1. Experimental P(Heart): $ \boxed{\frac{11}{36}} $
2. Theoretical P(Heart): $ \boxed{\frac{1}{4}} $
3. Experimental P(Diamond or Spade): $ \boxed{\frac{11}{18}} $
4. Theoretical P(Diamond or Spade): $ \boxed{\frac{1}{2}} $
5. Comparison: Experimental results vary slightly from theoretical due to small sample size; as trials increase, they should converge.

#### Eye Color Problems:

6a.
- P(blue) = $ \boxed{\frac{3}{20}} $
- P(brown) = $ \boxed{\frac{29}{40}} $
- P(green) = $ \boxed{\frac{1}{40}} $
- P(hazel) = $ \boxed{\frac{1}{10}} $

6b. P(blue or green) = $ \boxed{\frac{7}{40}} $

6c. P(not green or hazel) = $ \boxed{\frac{7}{8}} $

6d. Expected brown-eyed students: $ \boxed{261} $

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