Develop Non-Uniform Probability Models - 7th Grade Math
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Identify the observed relative frequency of an outcome with $15$ successes in $60$ trials.
Identify the observed relative frequency of an outcome with $15$ successes in $60$ trials.
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$\frac{15}{60}=0.25$. Relative frequency = successes ÷ total trials.
$\frac{15}{60}=0.25$. Relative frequency = successes ÷ total trials.
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What is a non-uniform probability model?
What is a non-uniform probability model?
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A model where outcomes have different probabilities. Unlike uniform models, probabilities vary between outcomes.
A model where outcomes have different probabilities. Unlike uniform models, probabilities vary between outcomes.
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Find the missing probability: $P(A)=0.35$, $P(B)=0.50$, so what is $P(C)$ if total is $1$?
Find the missing probability: $P(A)=0.35$, $P(B)=0.50$, so what is $P(C)$ if total is $1$?
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$0.15$. Since $0.35+0.50=0.85$, need $0.15$ to sum to $1$.
$0.15$. Since $0.35+0.50=0.85$, need $0.15$ to sum to $1$.
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Identify whether this model is valid: $P(A)=0.6$, $P(B)=0.5$ for two outcomes only.
Identify whether this model is valid: $P(A)=0.6$, $P(B)=0.5$ for two outcomes only.
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Not valid, because $0.6+0.5=1.1>1$. Probabilities exceed $1$, which is impossible.
Not valid, because $0.6+0.5=1.1>1$. Probabilities exceed $1$, which is impossible.
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Which option is the observed frequency for a model probability $0.25$ in $80$ trials (nearest whole)?
Which option is the observed frequency for a model probability $0.25$ in $80$ trials (nearest whole)?
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$0.25 times 80=20$. Multiply probability by trials to predict frequency.
$0.25 times 80=20$. Multiply probability by trials to predict frequency.
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What is the expected count for an outcome with model probability $0.30$ in $200$ trials?
What is the expected count for an outcome with model probability $0.30$ in $200$ trials?
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$0.30 times 200=60$. Expected count = probability × number of trials.
$0.30 times 200=60$. Expected count = probability × number of trials.
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Find and correct the error: A student says $P(A)= frac{7}{30}$ when $A$ occurred $7$ times in $20$ trials.
Find and correct the error: A student says $P(A)= frac{7}{30}$ when $A$ occurred $7$ times in $20$ trials.
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Correct: $P(A)= frac{7}{20}$. Should divide by actual trials ($20$), not $30$.
Correct: $P(A)= frac{7}{20}$. Should divide by actual trials ($20$), not $30$.
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Which option gives the model from data: outcome $X$ occurs $60$ times and $Y$ occurs $40$ times?
Which option gives the model from data: outcome $X$ occurs $60$ times and $Y$ occurs $40$ times?
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$P(X)=0.60$, $P(Y)=0.40$. Convert counts to probabilities: $ rac{60}{100}$ and $ rac{40}{100}$.
$P(X)=0.60$, $P(Y)=0.40$. Convert counts to probabilities: $ rac{60}{100}$ and $ rac{40}{100}$.
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What does it mean if observed frequencies are close to model probabilities as trials increase?
What does it mean if observed frequencies are close to model probabilities as trials increase?
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The model is reasonable; relative frequencies approach the model. Law of large numbers: frequencies converge to true probabilities.
The model is reasonable; relative frequencies approach the model. Law of large numbers: frequencies converge to true probabilities.
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Which option is the best model probability for an outcome seen $9$ times in $30$ trials?
Which option is the best model probability for an outcome seen $9$ times in $30$ trials?
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$\frac{9}{30}$. Simplifies to $\frac{3}{10}$ or $0.3$ as the probability.
$\frac{9}{30}$. Simplifies to $\frac{3}{10}$ or $0.3$ as the probability.
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Which option is closer to the observed probability $\frac{19}{40}$: $0.45$ or $0.50$?
Which option is closer to the observed probability $\frac{19}{40}$: $0.45$ or $0.50$?
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$0.50$, because $\frac{19}{40}=0.475$. $0.475$ is closer to $0.50$ than to $0.45$.
$0.50$, because $\frac{19}{40}=0.475$. $0.475$ is closer to $0.50$ than to $0.45$.
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Identify the model probabilities from frequencies: $R:5$, $G:3$, $B:2$ out of $10$ trials.
Identify the model probabilities from frequencies: $R:5$, $G:3$, $B:2$ out of $10$ trials.
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$P(R)=\frac{5}{10}$, $P(G)=\frac{3}{10}$, $P(B)=\frac{2}{10}$. Each probability = frequency \( \div \) total trials ($10$).
$P(R)=\frac{5}{10}$, $P(G)=\frac{3}{10}$, $P(B)=\frac{2}{10}$. Each probability = frequency \( \div \) total trials ($10$).
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Identify the experimental probability of rolling a $6$ if it occurs $14$ times in $90$ rolls.
Identify the experimental probability of rolling a $6$ if it occurs $14$ times in $90$ rolls.
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$\frac{14}{90}$. Divide occurrences by total rolls for probability.
$\frac{14}{90}$. Divide occurrences by total rolls for probability.
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Identify the probability model from observed frequencies: $A:12$, $B:8$ out of $20$ trials.
Identify the probability model from observed frequencies: $A:12$, $B:8$ out of $20$ trials.
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$P(A)=\frac{12}{20}$, $P(B)=\frac{8}{20}$. Divide each frequency by total trials to get probabilities.
$P(A)=\frac{12}{20}$, $P(B)=\frac{8}{20}$. Divide each frequency by total trials to get probabilities.
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Identify whether this model is valid: $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $P(C)=\frac{1}{6}$.
Identify whether this model is valid: $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $P(C)=\frac{1}{6}$.
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Valid, because $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$. Check: $\frac{3}{6}+\frac{2}{6}+\frac{1}{6}=\frac{6}{6}=1$ ✓
Valid, because $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$. Check: $\frac{3}{6}+\frac{2}{6}+\frac{1}{6}=\frac{6}{6}=1$ ✓
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What must the sum of all probabilities in a probability model equal?
What must the sum of all probabilities in a probability model equal?
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The sum must equal $1$. Total probability of all possible outcomes must be certain.
The sum must equal $1$. Total probability of all possible outcomes must be certain.
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What is a probability model for a chance process?
What is a probability model for a chance process?
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A list of outcomes with assigned probabilities summing to $1$. Each outcome gets a probability; all must total exactly $1$.
A list of outcomes with assigned probabilities summing to $1$. Each outcome gets a probability; all must total exactly $1$.
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What is the experimental probability of an outcome with frequency $f$ in $n$ trials?
What is the experimental probability of an outcome with frequency $f$ in $n$ trials?
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$p = \frac{f}{n}$. Experimental probability equals frequency divided by total trials.
$p = \frac{f}{n}$. Experimental probability equals frequency divided by total trials.
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What is the absolute difference between model $0.40$ and observed $\frac{18}{50}$?
What is the absolute difference between model $0.40$ and observed $\frac{18}{50}$?
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$\left|0.40 - \frac{18}{50}\right| = 0.04$. Calculate $|0.40-0.36|=0.04$ difference.
$\left|0.40 - \frac{18}{50}\right| = 0.04$. Calculate $|0.40-0.36|=0.04$ difference.
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Identify the experimental probability of heads if heads occurs $47$ times in $100$ flips.
Identify the experimental probability of heads if heads occurs $47$ times in $100$ flips.
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$\frac{47}{100}$. Divide favorable outcomes by total trials performed.
$\frac{47}{100}$. Divide favorable outcomes by total trials performed.
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Identify whether a model is uniform if outcomes have probabilities $0.10,0.20,0.70$.
Identify whether a model is uniform if outcomes have probabilities $0.10,0.20,0.70$.
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Non-uniform. Different probabilities mean non-uniform.
Non-uniform. Different probabilities mean non-uniform.
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A spinner landed on red $9$ times out of $30$. What model probability for red fits the data?
A spinner landed on red $9$ times out of $30$. What model probability for red fits the data?
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$
P(
ext{red})=
rac{9}{30}=
rac{3}{10}$. Use relative frequency as the model probability.
$ P( ext{red})= rac{9}{30}= rac{3}{10}$. Use relative frequency as the model probability.
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If the model says $P(E)=0.62$, what is $P(
ext{not }E)$?
If the model says $P(E)=0.62$, what is $P( ext{not }E)$?
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$1-0.62=0.38$. Subtract from $1$ to get complement.
$1-0.62=0.38$. Subtract from $1$ to get complement.
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A model predicts $P(A)=0.25$ in $200$ trials, but $A$ occurred $60$ times. What is observed $f$?
A model predicts $P(A)=0.25$ in $200$ trials, but $A$ occurred $60$ times. What is observed $f$?
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$f = \frac{60}{200} = 0.30$. Higher than expected; shows variation from model.
$f = \frac{60}{200} = 0.30$. Higher than expected; shows variation from model.
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A die roll experiment shows a $6$ occurred $25$ times in $150$ rolls. What is $P(6)$ from data?
A die roll experiment shows a $6$ occurred $25$ times in $150$ rolls. What is $P(6)$ from data?
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$P(6) = \frac{25}{150} = \frac{1}{6}$. Matches theoretical probability for a fair die.
$P(6) = \frac{25}{150} = \frac{1}{6}$. Matches theoretical probability for a fair die.
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