Approximate Probability From Collected Data - 7th Grade Math

1. Identify whether this is reasonable: with $p=\frac{1}{2}$ and $n=100$, predict exactly $50$ successes.

   Not guaranteed; $50$ is an estimate, not exact. Predictions give expected values, not exact guarantees.

2. Predict the approximate relative frequency of an event with $p=0.12$ over many trials.

   $0.12$. Relative frequency approaches the given probability value.

3. A bag has $5$ blue and $15$ total; predict blue draws in $90$ draws with replacement.

   $90\cdot\frac{5}{15}=30$. Expected blues: $90 \times \frac{5}{15} = 90 \times \frac{1}{3} = 30$.

4. Which value is the best estimate of probability from $7$ successes in $20$ trials?

   $\frac{7}{20}=0.35$. Best estimate uses the experimental probability formula $\frac{k}{n}$.

5. Which statement is correct: relative frequency will be exactly $p$ or will be close to $p$ in the long run?

   It will be close to $p$ in the long run. Results vary around $p$, not exactly equal.

6. What formula predicts the approximate relative frequency when probability is $p$?

   Approx. relative frequency $\approx p$. Relative frequency converges to probability in many trials.

7. What is the theoretical probability of rolling a $3$ on a fair number cube?

   $\frac{1}{6}$. One favorable outcome (3) out of six equally likely outcomes.

8. Identify the probability of rolling a $3$ or $6$ on a fair number cube.

   $\frac{2}{6}=\frac{1}{3}$. Two favorable outcomes ($3$ or $6$) out of six total.

9. What formula predicts the expected number of times an event occurs in $n$ trials?

   Expected count $= n \times p$. Multiply trials by probability to predict occurrences.

10. What is the predicted number of tails in $120$ flips of a fair coin?

    $120\times\frac{1}{2}=60$. Half the flips should be tails: $120 \times 0.5$.

11. Predict how many times a $3$ or $6$ appears in $600$ fair die rolls using $p=\frac{1}{3}$.

    $600\cdot\frac{1}{3}=200$. Expected value: multiply trials by combined probability $\frac{2}{6}=\frac{1}{3}$.

12. What is the predicted number of red outcomes in $50$ spins if $p(\text{red})=0.2$?

    $50\times^0.2=10$. Multiply spins by red probability: $50 \times 0.2$.

13. Find and correct the estimate: $p=\frac{1}{4}$ and $n=200$ gives $200\cdot\frac{1}{4}=40$.

    Correct: $200\cdot\frac{1}{4}=50$. Error found: $200 \times \frac{1}{4} = 50$, not 40.

14. What is the relative frequency of an event after $n$ trials with $k$ occurrences?

    $\frac{k}{n}$. Relative frequency equals occurrences divided by total trials.

15. If the experimental probability is $\frac{9}{50}$, what is the predicted count in $500$ trials?

    $500\cdot\frac{9}{50}=90$. Scale up: multiply trials by experimental probability.

16. What is the predicted number of $3$ or $6$ results in $600$ fair cube rolls?

    $600\times\frac{1}{3}=200$. Multiply trials by probability: $600 \times \frac{1}{3}$.

17. What is the predicted number of successes in $n$ trials for probability $p$?

    $n\cdot p$. Multiply trials by probability to predict expected successes.

18. If the relative frequency is $0.4$ over many trials, what probability should you estimate?

    $p\approx^0.4$. Long-run frequency approximates the true probability.

19. Identify the probability of rolling a $3$ on a fair number cube.

    $\frac{1}{6}$. One favorable outcome out of six equally likely outcomes.

20. What is the experimental probability of an event in $n$ trials with $k$ successes?

    $\frac{k}{n}$. Divide successes by total trials to get experimental probability.

21. Predict how many times a $6$ appears in $300$ fair die rolls using $p=\frac{1}{6}$.

    $300\cdot\frac{1}{6}=50$. Expected value: multiply trials by probability of rolling 6.

22. What happens to the typical variability of relative frequency as the number of trials increases?

    It decreases (results become more stable). More trials reduce random fluctuations in frequency.

23. Identify the formula to estimate probability from data: occurrences $k$, trials $n$.

    $\hat{p}=\frac{k}{n}$. Sample proportion formula: successes over trials estimates probability.

24. If the predicted count is $24$ in $80$ trials, what probability $p$ was used?

    $p=\frac{24}{80}=0.3$. Divide predicted count by trials: $\frac{24}{80}$.

25. What is the definition of relative frequency for an event $A$ in repeated trials?

    $\frac{\text{number of times }A\text{ occurs}}{\text{total number of trials}}$. Counts favorable outcomes divided by total trials.

26. If $p(A)=\frac{1}{5}$ and there are $150$ trials, what is the predicted count of $A$?

    $150\times\frac{1}{5}=30$. Multiply trials by probability: $150 \times \frac{1}{5}$.

27. What is the theoretical probability of rolling a $3$ or $6$ on a fair number cube?

    $\frac{2}{6}=\frac{1}{3}$. Two favorable outcomes (3 or 6) out of six equally likely outcomes.

28. If an event happens $45$ times in $100$ trials, what is the experimental probability?

    $\frac{45}{100}=0.45$. Experimental probability equals relative frequency.

29. What is the predicted number of heads in $80$ flips of a fair coin?

    $80\times\frac{1}{2}=40$. Half the flips should be heads: $80 \times 0.5$.

30. What is the relative frequency if an event occurs $7$ times in $20$ trials?

    $\frac{7}{20}=0.35$. Divide occurrences by total trials: $\frac{7}{20}$.

31. What does the long-run relative frequency of an event approach as trials increase?

    The probability of the event. Law of Large Numbers: frequency stabilizes to true probability.

32. Find the experimental probability if an event happens $18$ times in $60$ trials.

    $\frac{18}{60}=\frac{3}{10}$. Divide occurrences by trials: $\frac{18}{60}$ simplifies to $\frac{3}{10}$.

33. What is the predicted relative frequency for an event with probability $p$?

    $p$. Relative frequency equals the theoretical probability in the long run.

34. What does long-run relative frequency approach as the number of trials becomes large?

    It approaches the theoretical probability. Law of large numbers: experimental converges to theoretical probability.

35. Find the relative frequency if an event occurs $45$ times in $200$ trials.

    $\frac{45}{200}=0.225$. Divide occurrences by trials to get relative frequency as decimal.

36. Predict the number of heads in $80$ fair coin tosses using $p=\frac{1}{2}$.

    $80\cdot\frac{1}{2}=40$. Expected value: multiply trials by probability of heads.

37. If $p(A)=0.3$ and there are $200$ trials, what is the predicted count of $A$?

    $200\times^0.3=60$. Multiply trials by probability: $200 \times 0.3$.

38. What is the relative frequency if an event occurs $18$ times in $60$ trials?

    $\frac{18}{60}=0.3$. Divide occurrences by total trials: $\frac{18}{60}$.

39. A spinner has $8$ equal sections; predict hits on red in $160$ spins if $p=\frac{3}{8}$.

    $160\cdot\frac{3}{8}=60$. Expected hits: multiply spins by probability of red section.

40. If an event has probability $p=0.75$, what approximate relative frequency should you predict?

    $\approx^0.75$. Relative frequency approaches the given probability.

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