Cogito
Probability and Statistics · Chapter 3 · Lesson 2
Sample Spaces and Simulation
List everything that could happen.
12 problems · about 21 minutes · TEKS P.S.2.B, S-IC.A.2
What this lesson teaches
The student constructs sample spaces and designs simulations to estimate probabilities.
- A sample space lists every possible outcome.
- Order often matters, so HT and TH count separately.
- A simulation estimates a probability when counting is impractical.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThree coins. How many outcomes in the sample space?
Answer 8
Why 8.
Why is P(exactly one head on two coins) not one third?
Answer There are four outcomes, and two of them have one head.
Why Two of four outcomes qualify.
Three coins. How many outcomes?
Answer 8
Why 2 × 2 × 2.
Two coins. Probability of exactly one head, as a decimal?
Answer 0.5
Why 2 of 4.
Two dice. How many ways to total 7?
Answer 6
Why 1-6, 2-5, 3-4 and their reverses.
Build It Up
The same ideas with more to keep track of.
3 problemsTwo dice. Probability of a total of 7, as a decimal to three places?
Answer 0.167
Why 6 ÷ 36.
A coin then a die. How many outcomes?
Answer 12
Why 2 × 6.
Which gives a steadier simulation estimate? 1 for 20 trials, 2 for 2000.
Answer 2
Why More trials settle down.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the steps of designing a simulation in order.
Answer 1. State what one trial represents. 2. Decide how the random device maps onto the situation. 3. Define what counts as a success. 4. Run many trials and divide successes by trials.
Why You cannot count successes before defining one.
Four coins. How many outcomes?
Answer 16
Why 2 to the fourth.
The Two Coins: How many outcomes are in the sample space for two coins?
Answer 4
Why 4.
The Dice: How many outcomes for two dice?
Answer 36
Why 36.