Math worksheet titled 'Probability Worksheet 4' featuring problems on experimental and theoretical probability involving card suits and eye color data.

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 from a standard deck of 52 cards. The cards were drawn with replacement, so each draw is independent.

First, let’s count how many times each suit was selected using the tallies:

- Diamonds: `|||| ||` → 5 + 2 = 7
- Hearts: `|||| |||| |` → 5 + 4 + 1 = 10
- Spades: `|||| |||| |` → 5 + 5 + 1 = 11
- Clubs: `|||` → 3

Total number of trials = 7 + 10 + 11 + 3 = 31

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

$$
P_{\text{exp}}(\text{Heart}) = \frac{\text{Number of hearts drawn}}{\text{Total draws}} = \frac{10}{31}
$$

✔ Answer: $ \frac{10}{31} $

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

In a standard deck:
- There are 13 hearts out of 52 cards.

$$
P_{\text{theo}}(\text{Heart}) = \frac{13}{52} = \frac{1}{4} = 0.25
$$

✔ Answer: $ \frac{1}{4} $

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

Add the counts for diamonds and spades:
- Diamonds: 7
- Spades: 11
- Total = 7 + 11 = 18

$$
P_{\text{exp}}(\text{Diamond or Spade}) = \frac{18}{31}
$$

✔ Answer: $ \frac{18}{31} $

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

Each suit has 13 cards.

- Diamonds: 13
- Spades: 13
- Total favorable = 26

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

✔ Answer: $ \frac{1}{2} $

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

- Experimental Heart Probability: $ \frac{10}{31} \approx 0.3226 $
- Theoretical Heart Probability: $ \frac{1}{4} = 0.25 $
- Experimental is higher than theoretical.

- Experimental Diamond or Spade: $ \frac{18}{31} \approx 0.5806 $
- Theoretical Diamond or Spade: $ \frac{1}{2} = 0.5 $
- Experimental is higher than theoretical.

Findings:
- The experimental probabilities are close but not exactly equal to the theoretical ones.
- This is expected because the sample size (31 trials) is small, so random variation occurs.
- Over more trials, we would expect the experimental probabilities to get closer to the theoretical values due to the Law of Large Numbers.
- Hearts and black suits (spades/diamonds) were slightly overrepresented in this experiment.

✔ Answer: Experimental probabilities are somewhat higher than theoretical, likely due to sampling variability in a small number of trials.

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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{12}{80} = \frac{3}{20} $
- P(brown) = $ \frac{58}{80} = \frac{29}{40} $
- P(green) = $ \frac{2}{80} = \frac{1}{40} $
- P(hazel) = $ \frac{8}{80} = \frac{1}{10} $

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

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

✔ Answer: $ \frac{7}{40} $ or 0.175

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

That means they have blue or brown eyes.

$$
P(\text{not green or hazel}) = 1 - P(\text{green or hazel})
$$

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

So,
$$
P(\text{not green or hazel}) = 1 - \frac{1}{8} = \frac{7}{8}
$$

Alternatively:
- Blue + Brown = 12 + 58 = 70
- $ \frac{70}{80} = \frac{7}{8} $

✔ Answer: $ \frac{7}{8} $

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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 = $ 0.725 \times 360 $

$$
0.725 \times 360 = (7/10) \times 360 = 0.7 \times 360 = 252 \quad \text{(Wait, better calculate precisely)}
$$

$$
\frac{58}{80} = 0.725 \\
0.725 \times 360 = ?
$$

Break it down:
- $ 0.7 \times 360 = 252 $
- $ 0.025 \times 360 = 9 $
- So total = 252 + 9 = 261

✔ Answer: 261 students

Or:
$$
\frac{58}{80} \times 360 = \frac{58 \times 360}{80} = \frac{58 \times 9}{2} = \frac{522}{2} = 261
$$

✔ Final answer: 261

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✔ Final Answers Summary

#### Card Problems:
1. $ \frac{10}{31} $
2. $ \frac{1}{4} $
3. $ \frac{18}{31} $
4. $ \frac{1}{2} $
5. Experimental probabilities are slightly higher than theoretical; differences due to small sample size.

#### Eye Color Problems:
6a.
- P(blue) = $ \frac{12}{80} = \frac{3}{20} $
- P(brown) = $ \frac{58}{80} = \frac{29}{40} $
- P(green) = $ \frac{2}{80} = \frac{1}{40} $
- P(hazel) = $ \frac{8}{80} = \frac{1}{10} $

6b. $ \frac{14}{80} = \frac{7}{40} $

6c. $ \frac{70}{80} = \frac{7}{8} $

6d. 261 students

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