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

Let’s solve each part step by step.

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First, let’s understand Amanda’s card experiment.

She picked cards from a standard 52-card deck, recorded the suit, and replaced the card each time. The tally marks show how many times she got each suit:

- Diamonds: |||| | → that’s 5 + 2 = 7
- Hearts: |||| |||| → that’s 5 + 4 = 9
- Spades: |||| |||| | → that’s 5 + 5 + 1 = 11
- Clubs: ||| → that’s 3

Total trials = 7 + 9 + 11 + 3 = 30

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Question 1: Experimental probability of selecting a heart?

Experimental probability = (number of times heart occurred) / (total trials)

= 9 / 30

Simplify: divide numerator and denominator by 3 → 3/10

✔ Answer: 3/10

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Question 2: Theoretical probability of selecting a heart?

In a standard deck, there are 4 suits, each with 13 cards.

So, theoretical probability = 13 hearts / 52 total cards = 1/4

✔ Answer: 1/4

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Question 3: Experimental probability of selecting a diamond or a spade?

Diamonds: 7
Spades: 11
Total for diamond OR spade = 7 + 11 = 18

Total trials = 30

Probability = 18 / 30 = simplify by dividing by 6 → 3/5

✔ Answer: 3/5

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Question 4: Theoretical probability of selecting a diamond or a spade?

Diamonds: 13 cards
Spades: 13 cards
Total favorable = 13 + 13 = 26

Total cards = 52

Probability = 26 / 52 = 1/2

✔ Answer: 1/2

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Question 5: Compare results and describe findings.

Experimental vs Theoretical:

- Heart: Experimental = 3/10 (0.3), Theoretical = 1/4 (0.25) → close, but experimental is a bit higher.
- Diamond or Spade: Experimental = 3/5 (0.6), Theoretical = 1/2 (0.5) → again, experimental is higher.

This makes sense because in small experiments (only 30 trials), results can vary from theory due to chance. As you do more trials, experimental probability usually gets closer to theoretical.

✔ Answer: The experimental probabilities are close to the theoretical ones but not exact — this is normal with small sample sizes. With more trials, they’d likely get even closer.

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Now, Dale’s eye color survey:

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

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

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Question 6a: Experimental probability distribution

P(blue) = 12 / 80 = simplify → divide by 4 → 3/20

P(brown) = 58 / 80 = simplify → divide by 2 → 29/40

P(green) = 2 / 80 = 1/40

P(hazel) = 8 / 80 = 1/10

✔ Answers:
- P(blue) = 3/20
- P(brown) = 29/40
- P(green) = 1/40
- P(hazel) = 1/10

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Question 6b: Probability student has blue OR green eyes?

Blue: 12, Green: 2 → total = 14

Probability = 14 / 80 = simplify → divide by 2 → 7/40

✔ Answer: 7/40

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Question 6c: Probability student does NOT have green or hazel eyes?

Green + Hazel = 2 + 8 = 10

So, NOT green or hazel = Total - 10 = 80 - 10 = 70

Probability = 70 / 80 = simplify → divide by 10 → 7/8

✔ Answer: 7/8

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Question 6d: If grade has 360 students, how many expected to have brown eyes?

From survey, P(brown) = 58/80 = 29/40

Expected number = (29/40) × 360

Calculate:

First, 360 ÷ 40 = 9
Then, 29 × 9 = 261

✔ Answer: 261

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Final Answer:

1. 3/10
2. 1/4
3. 3/5
4. 1/2
5. The experimental probabilities are close to the theoretical ones but not exact — this is normal with small sample sizes. With more trials, they’d likely get even closer.
6a. P(blue) = 3/20, P(brown) = 29/40, P(green) = 1/40, P(hazel) = 1/10
6b. 7/40
6c. 7/8
6d. 261