Warm-up
Three quick questions from earlier lessons.
Discover
A fair coin gave 7 heads in 10 tosses. If you toss it 10,000 times, what do you expect?
- A proportion of heads close to 0.5
- More tails than heads, to make up for the 7 heads
Run 10 tosses a few times, then 1,000. What happens to the proportion of heads?
- It gets close to 0.5
- It swings further from 0.5 to make up for streaks
Key idea
Probability is a long-run relative frequency.
Law of large numbers: as a random process is repeated many times, the proportion of times an event occurs gets close to its probability. Independent trials have no memory. A simulation imitates a random process with random numbers to estimate probabilities.
Worked example
Simulate a 30% chance with random digits.
Let 0, 1, 2 mean "success"
Each digit has probability 0.1.
Read 1,000 random digits; count successes
Repeat many times.
About 300 successes expected
Relative frequency.
Your turn, guided
Your turn: how many digits for a 60% chance?
0.6 ÷ 0.1.
Practice ★
- A fair coin lands heads 7 times in a row. What is the probability of heads on the next toss?
- 1/2: the coin has no memory
- Less than 1/2: tails is due
- More than 1/2: heads is on a streak
- What does it mean that the probability of rain tomorrow is 0.7?
- On many days like tomorrow, it rains on about 70% of them
- It will rain for 70% of the day
- It will rain if the probability is above 0.5
Watch out!
Practice ★★
- To simulate a free throw made with probability 0.6 using random digits 0–9, how many of the ten digits should stand for 'made'?
- To simulate a free throw made with probability 0.4 using random digits 0–9, how many of the ten digits should stand for 'made'?
Exit ticket
Three questions to check you have got it.
- A fair coin lands heads 7 times in a row. What is the probability of heads on the next toss?
- 1/2: the coin has no memory
- Less than 1/2: tails is due
- More than 1/2: heads is on a streak
- A fair coin lands heads 5 times in a row. What is the probability of heads on the next toss?
- 1/2: the coin has no memory
- Less than 1/2: tails is due
- More than 1/2: heads is on a streak
- A fair coin lands heads 10 times in a row. What is the probability of heads on the next toss?
- 1/2: the coin has no memory
- Less than 1/2: tails is due
- More than 1/2: heads is on a streak
Summary card
Long run
- Probability = long-run relative frequency.
- Simulate with random numbers.
Watch out: no memory, no "due" outcomes.