Probability Models: Uniform and Non-Uniform

Which set correctly represents the complete sample space when rolling a standard six-sided die?

A probability model has exactly three outcomes: A, B, and C. The model assigns $P(A) = 0.35$ and $P(B) = 0.25$. All probabilities must sum to 1. What is $P(C)$?

A die is rolled 60 times. Face 4 appears 12 times. Express the relative frequency of face 4 as a decimal.

Which of the following is a valid probability model for a spinner with three colors?

A fair 8-sided die has faces numbered 1 through 8. All outcomes are equally likely. Using a uniform probability model, compute the probability of rolling a number greater than 5. Express your answer as a fraction.

A bag contains 4 red tiles, 6 blue tiles, and 2 green tiles. One tile is selected at random; every tile is equally likely to be chosen. Using a uniform probability model, compute $P(\text{blue})$ as a fraction in lowest terms.

A spinner is spun 80 times. Results: Red = 28, Blue = 32, Yellow = 20. Build a non-uniform probability model from this data. What is $P(\text{Yellow})$ expressed as a decimal?

A non-uniform spinner model assigns $P(\text{Red}) = 0.35$, $P(\text{Blue}) = 0.40$, and $P(\text{Yellow}) = 0.25$. What is the probability of landing on Red or Blue?

A student rolls a 4-sided die with faces labeled A, B, C, and D, and writes $P(A) = \frac{1}{4}$. When is this a valid application of the uniform probability model?

A probability model predicts that Red will appear 30 times in 100 spins. In an actual experiment, Red appears 27 times. A student claims the model is wrong because the prediction did not match. Is the student correct? Explain your reasoning.

A student creates a probability model for rolling a standard six-sided die: $P(1) = P(2) = P(3) = P(4) = 0.25$. What is the error in this model?

The table shows two experiments using the same spinner model ($P(\text{Red}) = 0.5$, $P(\text{Blue}) = 0.3$, $P(\text{Green}) = 0.2$) compared to observed data from 100 spins each. In which scenario does the model fit better, and why?

Use the frequency data to answer both parts below.

A special card deck contains only 8 red cards, 4 blue cards, and 4 green cards. Every card is equally likely to be drawn. Using a uniform probability model, compute the probability of drawing a card that is not red. Express your answer as a fraction in lowest terms.

A student draws 20 marbles from a bag with replacement and records: Red = 3, Blue = 12, Yellow = 5. She builds a non-uniform probability model: $P(\text{Red}) = 0.15$, $P(\text{Blue}) = 0.60$, $P(\text{Yellow}) = 0.25$. She says she is very confident this model accurately represents the bag. Is this confidence justified? Explain.

A spinner model predicts $P(\text{Red}) = 0.40$, $P(\text{Blue}) = 0.35$, $P(\text{Green}) = 0.25$. A student spins 200 times and gets: Red = 78, Blue = 71, Green = 51. The predicted frequencies are: Red = 80, Blue = 70, Green = 50. Which conclusion is best supported?

What error did the student make?

What error did the student make?

A spinner is spun 400 times with results: Red = 180, Blue = 110, Green = 70, Yellow = 40.
(a) Build a non-uniform probability model. Express each probability as a decimal.
(b) Verify that the four probabilities sum to 1.
(c) Using your model, find $P(\text{Red or Green})$.
(d) A second spinner has 4 equal sections labeled Red, Blue, Green, and Yellow. For that spinner's uniform model, what is $P(\text{Red or Green})$? Why do the two models give different answers?

A probability model for a 4-color spinner: $P(\text{Red}) = 0.30$, $P(\text{Blue}) = 0.40$, $P(\text{Green}) = 0.20$, $P(\text{Yellow}) = 0.10$.
In an experiment with 200 spins, the results are: Red = 52, Blue = 83, Green = 42, Yellow = 23.
(a) Compute the predicted frequency for each color (use $P \times 200$).
(b) Find the difference (observed − predicted) for each color.
(c) Do these data suggest the model is accurate? Justify your answer by discussing which colors fit well and which may be cause for concern.