Probability 1
A bag contains 3 red marbles, 5 blue marbles, and 2 yellow marbles.
One marble is drawn at random and not replaced. Then a second marble is drawn.
What is the probability that both marbles are blue?
2/9
Explain what it means for outcomes to be equally likely and give a classroom example
Equally likely means each outcome has the same probability. Example: spinning a fair 4-color spinner with equal sectors — each color has probability 1/4
Define relative frequency and explain how it relates to probability in simple terms.
Relative frequency = number of times an event occurs divided by total number of trials. It estimates the probability of the event; with many trials it tends to approach the theoretical probability
Identify whether the probability value 00, 0.250.25, 0.50.5, 0.750.75, and 11 correspond to impossible, unlikely, equally likely, likely, or certain.
What is the term for a part of a population that is used to make conclusions about the whole population?
Sample
If you roll a fair six-sided die, what is the probability of rolling an even number? Express as a fraction, decimal, and word description (likely/unlikely/etc.)
Even outcomes = {2, 4, 6}, so probability = 3/6 =1/2 = 0.5. Description: equally likely / neither unlikely nor likely (around 1/2).
A jar contains 3 red, 2 blue, and 1 green marble. Are outcomes equally likely if you draw one marble? Explain and compute the probability of drawing a blue marble.
Outcomes (each marble) are equally likely if each physical marble is equally likely to be drawn. Probability(draw blue) = number blue / total = 2/6= 1/3 = 0.333
A fair coin is flipped 100 times and lands heads 53 times. What is the observed relative frequency of heads? How does it compare to the theoretical probability?
Observed relative frequency = 53/100 = 0.53.
Theoretical probability of heads = 1/2=0.5. Comparison: observed is close; slight discrepancy due to chance variation.
Classify the event "rolling a 7 on a single six-sided die" as impossible, unlikely, equally likely, likely, or certain, and explain.
Impossible, because no face shows 7; probability 00
Define "population" in a statistics context and give a simple classroom example.
the full set of individuals or items of interest (e.g., all students in a school).
A spinner has 4 equal sections colored red, blue, green, and yellow. If you spin once, what is the probability of landing on either red or blue? Show the sample space and your reasoning.
Sample space = {red, blue, green, yellow}. Probability(red or blue) = 2/4 = 1/2 = 0.5
Use a tree diagram or organized list to find the sample space for tossing two fair coins (show all the options of how the coins could land). Then find probability of landing on heads.
Sample space = {HH, HT, TH, TT}.
Exactly one head outcomes = {HT, TH} → probability = 2/4 = 1/2= 0.5. Then find probability of landing on head.
A spinner has 4 equal sections labeled A, B, C, and D.
Theoretical probability of landing on A is 1/4.
After 40 spins, A landed 18 times.
Is the experimental probability close to the theoretical probability? Explain.
No.
Experimental probability = 18/40 = 0.45
Theoretical probability = 0.25
These are not close.
Two models give different probabilities for an event: Model A predicts 0.6, Model B predicts 0.4. You observe the event 70 out of 100 times. Which model matches the observed frequency better?
Model A is closer to the observed frequency.
A principal wants to know students’ favorite lunch. She surveys 15 randomly selected students from the entire school. What is the population?
All the students in the school
You draw one card at random from a deck of 4 cards labeled A, B, C, D where each is equally likely. Write a probability model (list outcomes and probability for each). Then find the probability of drawing a vowel if E was added to the deck.
Original model: P(A)=P(B)=P(C)=P(D)= 1/4.
If deck is {A, B, C, E} and assuming all equally likely, vowel outcomes = {A, E} so P(vowel) = 2/5
A fair six-sided die is rolled twice. What is the probability the sum of the two rolls is 7? Show the sample space or an organized method to compute
Ordered pairs 36 outcomes. Pairs summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 outcomes. Probability = 6/36 = 1/6 = ≈ 0.1667
You simulate spinning a fair spinner 50 times and get 20 red, 15 blue, 10 green, 5 yellow. Compute observed relative frequencies and compare them to the model probabilities if the spinner is fair. Evaluate the level of agreement.
The test scores are:
95, 72, 80, 75, 78
What is the median?
78
A survey about school lunches is given only to students who bring their lunch from home. Is this sample likely to be representative? Explain.
No, because it is a biased sample.
(It does not represent students who buy lunch.)
The probability of choosing a weekday out of the days of the week. Identify whether the choosing Saturday is unlikely, equally likely, or likely.
5/7
unlikely
A bag contains:
5 red marbles
4 blue marbles
3 green marbles
Two marbles are drawn without replacement.
What is the probability that the first marble is red and the second marble is green
15/132 or 5/144
Explain how increasing the number of trials in a chance experiment affects the relationship between observed relative frequency and theoretical probability. Use an example (coin flips or die rolls) and explain why.
As number of trials increases, observed relative frequency tends to get closer to the theoretical probability (Law of Large Numbers). Example: flipping a fair coin 10 times might give 7 heads (0.7), but flipping 10,000 times is likely to produce a relative frequency closer to 0.50.5. Larger samples reduce random variation.
A company wants to know how adults in a city feel about public transportation. They survey only people waiting at a bus stop.
Why might this sample not be representative?
Because it only includes people who already use public transportation, so it is biased
Explain why random sampling tends to produce representative samples. Give one example of a nonrandom sampling method and why it might be biased
Random sampling gives every member an equal chance of selection, reducing systematic bias and increasing likelihood the sample reflects population characteristics. Nonrandom example: surveying only students in the front row — biased because seating may correlate with behavior or demographics.