A sample space is the list of every possible outcome. Two coins give four outcomes, not three.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Order can matter
Head-then-tail and tail-then-head are different outcomes. Merging them is why "one of each" gets miscounted as one third.
Tree diagrams
A tree branches once per stage. Following a path from root to tip gives one complete outcome.
Simulation
When counting is hard, imitate the situation with dice or random digits and run it many times.
Designing one
State what one trial is, how the randomness maps onto the situation, and what counts as a success.
How many trials
More trials give a steadier estimate. Ten trials tell you very little; a thousand tell you a great deal.
List everything that could happen
The sample space is the set of all possible outcomes. Writing it out — with a tree diagram or a table for two-stage processes — turns a confusing probability question into a counting exercise.
Be careful what counts as an outcome
Rolling two dice has 36 equally likely outcomes, not 11 sums. Treating the sums as equally likely is the classic error, and listing the sample space properly is what exposes it.
Simulation when theory is hard
Repeat a random process many times and count how often the event occurs. The result estimates the probability, and it works for situations far too complicated to compute directly.
Designing a simulation
State how randomness is generated, what one trial is, what you record, and how many trials you run. Without those four, no one can reproduce or evaluate the result.
Step 2: Try It Yourself
Tap and try it out.
HH
HT
TH
TT
Step 3: Watch an Example
One step at a time.
Watch Omar List a Sample Space
Omar tosses two coins and wants the probability of exactly one head.
- Step 1
He lists all four outcomes: HH, HT, TH and TT.
Step 4: Your Turn
Practice makes it stick.
The Two Coins
Problem 1 of 2
How many outcomes are in the sample space for two coins?
The Dice
Problem 2 of 2
How many outcomes for two dice?
List and Simulate
1 of 8
Three coins. How many outcomes?
2 of 8
Two coins. Probability of exactly one head, as a decimal?
3 of 8
Two dice. How many ways to total 7?
4 of 8
Two dice. Probability of a total of 7, as a decimal to three places?
5 of 8
A coin then a die. How many outcomes?
6 of 8
Which gives a steadier simulation estimate? 1 for 20 trials, 2 for 2000.
7 of 8
Put the steps of designing a simulation in order.
- 1Decide how the random device maps onto the situation.
- 2Define what counts as a success.
- 3Run many trials and divide successes by trials.
- 4State what one trial represents.
8 of 8
Four coins. How many outcomes?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Three coins. How many outcomes in the sample space?
Question 2 of 2
Why is P(exactly one head on two coins) not one third?
What You Learned
- 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.