Develop Non-Uniform Probability Models - 7th Grade Math

1. Which option describes how to build a probability model from observed frequencies?

   Divide each outcome count by the total number of trials. Relative frequency becomes the probability in the model.

2. What is the absolute difference between model $0.40$ and observed $\frac{18}{50}$?

   $\left|0.40 - \frac{18}{50}\right| = 0.04$. Calculate $|0.40-0.36|=0.04$ difference.

3. What is the expected count for an outcome with probability $p$ in $n$ trials?

   $ \text{expected count}= np$. Multiply trials by probability to predict occurrences.

4. Which option is the observed frequency for a model probability $0.25$ in $80$ trials (nearest whole)?

   $0.25times 80=20$. Multiply probability by trials to predict frequency.

5. What is the experimental probability of an outcome with frequency $f$ in $n$ trials?

   $p = \frac{f}{n}$. Experimental probability equals frequency divided by total trials.

6. What is the predicted count for outcome $A$ if the model gives $P(A)=0.4$ in $n=50$ trials?

   $0.4\times 50=20$. Expected count is probability times number of trials.

7. Identify the model probability from frequencies: $R:12$, $B:8$ out of $20$ total. What is $P(B)$?

   $\frac{8}{20}=\frac{2}{5}$. Blue occurred 8 times out of 20 total trials.

8. Which option is closer to the observed probability $\frac{19}{40}$: $0.45$ or $0.50$?

   $0.50$, because $\frac{19}{40}=0.475$. $0.475$ is closer to $0.50$ than to $0.45$.

9. What is the absolute difference between model and observed: model $P(A)=0.30$, observed $\frac{18}{50}$?

   $\left|0.30-0.36\right|=0.06$. Find $|0.30 - 0.36|$ where $\frac{18}{50} = 0.36$.

10. Identify whether this is uniform: outcomes $1$-$4$ have probabilities $0.10,0.20,0.30,0.40$.

    Non-uniform. Probabilities differ, so outcomes are not equally likely.

11. What does it mean if observed frequencies are close to model probabilities as trials increase?

    The model is reasonable; relative frequencies approach the model. Law of large numbers: frequencies converge to true probabilities.

12. Find the missing probability: $P(A)=0.35$, $P(B)=0.50$, so what is $P(C)$ if total is $1$?

    $0.15$. Since $0.35+0.50=0.85$, need $0.15$ to sum to $1$.

13. Identify the missing probability so the model sums to $1$: $P(A)=\frac{1}{6}$, $P(B)=\frac{1}{2}$, $P(C)=?$

    $\frac{1}{3}$. Since $\frac{1}{6} + \frac{1}{2} = \frac{2}{3}$, need $\frac{1}{3}$ to sum to 1.

14. What is the difference between theoretical probability and experimental probability?

    Theoretical uses a model; experimental uses observed frequency. Theoretical is based on assumptions; experimental is based on actual data.

15. Identify whether this model is valid: $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $P(C)=\frac{1}{6}$.

    Valid, because $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$. Check: $\frac{3}{6}+\frac{2}{6}+\frac{1}{6}=\frac{6}{6}=1$ ✓

16. Which is closer to $f=0.41$: $p_1=0.38$ or $p_2=0.46$?

    $p_1=0.38$. $|0.38-0.41|=0.03$ is less than $|0.46-0.41|=0.05$.

17. What is the predicted count for outcome $B$ if the model gives $P(B)=\frac{3}{10}$ in $n=60$ trials?

    $\frac{3}{10}\times 60=18$. Expected count is probability times number of trials.

18. What is the observed relative frequency of success for $k=9$ successes in $n=30$ trials?

    $\frac{9}{30}=\frac{3}{10}$. Success rate is successes divided by total trials.

19. What is the complementary probability formula for an event $E$ in a model?

    $P(\text{not } E)=1-P(E)$. Complement rule: probabilities sum to $1$.

20. Find and correct the error: A student says $P(A)=frac{7}{30}$ when $A$ occurred $7$ times in $20$ trials.

    Correct: $P(A)=frac{7}{20}$. Should divide by actual trials ($20$), not $30$.

21. Which option gives the model from data: outcome $X$ occurs $60$ times and $Y$ occurs $40$ times?

    $P(X)=0.60$, $P(Y)=0.40$. Convert counts to probabilities: $\frac{60}{100}$ and $\frac{40}{100}$.

22. What is a probability model for a chance process?

    A list of outcomes with assigned probabilities summing to $1$. Each outcome gets a probability; all must total exactly $1$.

23. A die roll experiment shows a $6$ occurred $25$ times in $150$ rolls. What is $P(6)$ from data?

    $P(6) = \frac{25}{150} = \frac{1}{6}$. Matches theoretical probability for a fair die.

24. Find the expected number of heads in $40$ flips if the model uses $P(H) = \frac{1}{2}$.

    $40 \frac{1}{2} = 20$. Half the flips should be heads if fair.

25. What is the relative frequency of an outcome observed $k$ times in $n$ trials?

    $\frac{k}{n}$. Count of occurrences divided by total trials.

26. What must be true about the probabilities in a probability model?

    Each is between $0$ and $1$, and totals sum to $1$. Probabilities must be valid and exhaustive.

27. Compare model to data: model $P(A)=\frac{1}{2}$; observed $A$ occurred $27$ times in $50$ trials. Is observed higher or lower?

    Higher, since $\frac{27}{50}=0.54>0.5$. Observed frequency 0.54 exceeds model probability 0.5.

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

    $P(E) = \frac{k}{n}$. Divide successful outcomes by total trials.

29. Find $P(H)$ from $50$ coin flips with $H:28$ and $T:22$ using a frequency model.

    $\frac{28}{50}=\frac{14}{25}$. Heads occurred 28 times out of 50 flips.

30. Identify the model probability from frequencies: $R:12$, $B:8$ out of $20$ total. What is $P(R)$?

    $\frac{12}{20}=\frac{3}{5}$. Red occurred 12 times out of 20 total trials.

31. Which option is the best model probability for an outcome seen $9$ times in $30$ trials?

    $\frac{9}{30}$. Simplifies to $\frac{3}{10}$ or $0.3$ as the probability.

32. What must be true about the probabilities in any valid probability model?

    Each is $\ge 0$ and the total sum is $1$. Probabilities can't be negative and must sum to certainty.

33. A model predicts $P(A)=0.25$ in $200$ trials, but $A$ occurred $60$ times. What is observed $f$?

    $f = \frac{60}{200} = 0.30$. Higher than expected; shows variation from model.

34. Find the expected number of times outcome $A$ occurs in $80$ trials if $P(A)=0.15$.

    $ 80(0.15)=12$. Multiply total trials by the probability.

35. What happens to experimental probabilities as the number of trials becomes very large?

    They tend to approach the true (model) probability. Law of large numbers in action.

36. Compare $p=0.30$ to $f=0.27$: what is $|p-f|$?

    $|0.30-0.27|=0.03$. Find absolute value of the difference.

37. What does it mean for a probability model to be non-uniform?

    Outcomes have different probabilities, not all equal. Some outcomes are more likely than others.

38. Identify the observed relative frequency of an outcome with $15$ successes in $60$ trials.

    $\frac{15}{60}=0.25$. Relative frequency = successes ÷ total trials.

39. What is a non-uniform probability model?

    A model where outcomes have different probabilities. Unlike uniform models, probabilities vary between outcomes.

40. Identify whether a model is uniform if outcomes have probabilities $0.10,0.20,0.70$.

    Non-uniform. Different probabilities mean non-uniform.

41. Identify the model probability if an outcome occurred $18$ times in $60$ trials.

    $ P= \frac{18}{60}= \frac{3}{10}$. Simplify the fraction of successes to trials.

42. Identify the experimental probability of heads if heads occurs $47$ times in $100$ flips.

    $\frac{47}{100}$. Divide favorable outcomes by total trials performed.

43. Identify the observed frequency (as a fraction) if an event happened $7$ times in $50$ trials.

    $f=\frac{7}{50}$. Express occurrences as a fraction of total trials.

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

    $P(\text{outcome})=\frac{k}{n}$. Divide the number of successes by the total number of trials.

45. What is the expected count for an outcome with model probability $0.30$ in $200$ trials?

    $0.30times 200=60$. Expected count = probability × number of trials.

46. Identify the experimental probability of rolling a $6$ if it occurs $14$ times in $90$ rolls.

    $\frac{14}{90}$. Divide occurrences by total rolls for probability.

47. What is the absolute difference between model probability $p$ and observed frequency $f$?

    $|p-f|$. Measures how close observed is to predicted.

48. Find $P(T)$ from $50$ coin flips with $H:28$ and $T:22$ using a frequency model.

    $\frac{22}{50}=\frac{11}{25}$. Tails occurred 22 times out of 50 flips.

49. Identify the probability model from observed frequencies: $A:12$, $B:8$ out of $20$ trials.

    $P(A)=\frac{12}{20}$, $P(B)=\frac{8}{20}$. Divide each frequency by total trials to get probabilities.

50. Which option is the best probability model from $40$ trials: $A:10$, $B:20$, $C:10$?

    $P(A)=\frac{1}{4},\ P(B)=\frac{1}{2},\ P(C)=\frac{1}{4}$. Each outcome's probability equals its frequency divided by 40.

51. A spinner landed on red $9$ times out of $30$. What model probability for red fits the data?

    $ P($\text{red}$)= \frac{9}{30}= \frac{3}{10}$. Use relative frequency as the model probability.

52. Identify the model probabilities from frequencies: $R:5$, $G:3$, $B:2$ out of $10$ trials.

    $P(R)=\frac{5}{10}$, $P(G)=\frac{3}{10}$, $P(B)=\frac{2}{10}$. Each probability = frequency $\div$ total trials ($10$).

53. What must the sum of all probabilities in a probability model equal?

    The sum must equal $1$. Total probability of all possible outcomes must be certain.

54. If the model says $P(E)=0.62$, what is $P( \text{not }E)$?

    $1-0.62=0.38$. Subtract from $1$ to get complement.

55. Identify the missing probability so the model sums to $1$: $P(A)=0.2$, $P(B)=0.5$, $P(C)=?$

    $0.3$. Sum of all probabilities must equal 1, so $1 - 0.2 - 0.5 = 0.3$.

56. Identify whether this is uniform: outcomes $1$-$4$ have probabilities $0.25,0.25,0.25,0.25$.

    Uniform. All outcomes have equal probability of $\frac{1}{4}$.

57. Observed $f(\text{win}) = \frac{11}{20}$. What decimal probability model matches this frequency?

    $\frac{11}{20} = 0.55$. Convert fraction to decimal form.

58. What does it mean to develop a probability model from observed frequencies?

    Assign probabilities using long-run relative frequencies. Use observed data to estimate probabilities.

59. Identify whether this model is valid: $P(A)=0.6$, $P(B)=0.5$ for two outcomes only.

    Not valid, because $0.6+0.5=1.1>1$. Probabilities exceed $1$, which is impossible.

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