Probability Models
Create and use models to predict outcomes of random events.
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Probability Models
- Sample space: List of all possible outcomes
- Tree diagrams: Visual representation of multi-step experiments
- Tables: Organized display of compound event outcomes
- Simulations: Using random processes to model real situations
Practice
Question 1: List the sample space for flipping a coin twice.
Answer
Sample space: {HH, HT, TH, TT} - 4 outcomes.
Question 2: How many outcomes are in the sample space for rolling two dice?
Answer
6 x 6 = 36 outcomes in the sample space.
Question 3: A spinner has red, blue, yellow. You spin twice. How many outcomes in sample space?
Answer
3 x 3 = 9 outcomes: RR, RB, RY, BR, BB, BY, YR, YB, YY.
Question 4: Using a tree diagram, how many branches would flipping 3 coins have?
Answer
2 x 2 x 2 = 8 final branches representing 8 outcomes.
Question 5: A restaurant offers 3 entrees and 4 desserts. How many meal combinations?
Answer
3 x 4 = 12 combinations in the sample space.
Question 6: To simulate a 70% chance of rain, which model works: roll 1-7 on 10-sided die means rain?
Answer
Yes, rolling 1-7 out of 10 gives P = 7/10 = 70%.
Question 7: How could you simulate a fair coin using a 6-sided die?
Answer
Roll 1-3 = heads, roll 4-6 = tails. Each has P = 3/6 = 1/2.
Question 8: In a simulation of 50 coin flips, you got 28 heads. What is the experimental probability?
Answer
P(heads) = 28/50 = 0.56 or 56%.
Question 9: You want to simulate a 25% chance. What could you use?
Answer
Roll a die: 1 or 2 = success (but this gives 2/6 = 33%). Better: use cards A-4 and draw a specific card, or use spinner with 1/4 section.
Question 10: A simulation ran 100 trials with 23 successes. What probability does this estimate?
Answer
Estimated probability = 23/100 = 0.23 or 23%.