Year 10 Standard · Unit 4 Sets, Venn diagrams and probability · Lesson 6 of 6

What does a probability promise about a short run of events?

A probability says how likely one result is. This lesson tests what that does, and does not, tell you about 50 flips of a coin, and about 1000.

Statistics and ProbabilityGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 3: a fair die is rolled. What is P(a prime number)?
    Answer1/2. The primes are 2, 3 and 5: 3 of the 6 outcomes.
  2. From Year 8: 0.35 × 200 = ?
    Answer70. 0.35 × 200 = 35 × 2.
  3. From Lesson 4: a fair coin is tossed twice. What is P(at least one head)?
    Answer3/4. 1 − P(no heads) = 1 − 1/4.
Words used in this lesson
trial
one repeat of an experiment. For example: one spin of a spinner.
relative frequency
the number of times an event happens divided by the number of trials. For example: 7 heads in 20 flips is 0.35.
experimental probability
a probability estimated from results. For example: a cup landing upright in 13 of 50 throws.
theoretical probability
a probability worked out from equally likely outcomes. For example: 1/2 for a fair coin.
expected frequency
the probability multiplied by the number of trials. For example: 0.2 × 50 trials.
random
unpredictable one trial at a time, but with a pattern in the long run. For example: the face of a fair die.

Made-up flips and real flips

A teacher in Wellington asks half the class to flip a fair coin 50 times and record the results, and the other half to write down a list of 50 flips that they make up. She leaves the room, comes back, and picks out the real list from the invented one in a few seconds, and she does it every year.

Predict first: what do you think she is looking for? In a real list of 50 fair flips, how long a run of the same face in a row would you expect to see at least once, and does a made-up list tend to have longer or shorter runs? Then use the first figure.

1
25heads in 50 flips
4longest run in a row

Sequence 1: 25 heads and 25 tails in 50 flips. The longest run is 4 heads in a row.

Show me the steps
  1. Sequence 1: 25 heads, and the longest run is 4 heads in a row.
  2. Slide to sequence 3: only 20 heads out of 50, and a run of 7 tails in a row at the start. Both happen with a fair coin.
  3. Slide to sequence 4: a run of 7 tails again, with 23 heads. Long runs are not rare.
  4. Slide to sequence 5: 25 heads and 25 tails exactly, yet the longest run is 5. An even split does not mean short runs.

Random does not mean evenly spread, and the proportion settles with many trials.

  • Experimental probability (relative frequency) = number of times the event happened ÷ number of trials. It is an estimate.
  • Theoretical probability comes from equally likely outcomes. A fair coin: P(heads) = 1/2.
  • More trials give a more reliable estimate, because the relative frequency settles near the true probability.
  • Expected frequency = P(event) × number of trials. It is what you expect on average, not what must happen.
  • Each trial is independent: earlier results do not change the next one.
10
0.6run A, relative frequency
0.2gap between the two runs

After 10 flips: run A has 6 heads (0.6) and run B has 4 heads (0.4). The runs are 0.2 apart.

Show me the steps
  1. After 10 flips: run A has 6 heads (0.6) and run B has 4 heads (0.4). The runs are 0.2 apart.
  2. Slide to 100 flips: run A has 61 heads (0.61) and run B has 42 heads (0.42). Still 0.19 apart: 100 flips is not yet many.
  3. Slide to 500 flips: run A has 250 heads (0.5) and run B has 236 heads (0.472), a gap of 0.028.
  4. Slide to 1000 flips: run A has 477 heads (0.477) and run B has 493 heads (0.493), a gap of 0.016. Both are close to 0.5, but neither is exactly 0.5.

There is no “due”. A coin has no memory, so a long run does not change what comes next. The proportion settles because later flips make up most of a large total, not because heads have to catch up. A run of 5 or more in a row happens in about 4 out of 5 sets of 50 flips.

Worked example: estimate and expect

A drawing pin is dropped 150 times and lands point up 93 times. Estimate P(point up), then estimate how many of 400 drops will land point up.

  1. Relative frequency: 93 ÷ 150 = 0.62. This is the experimental probability.
  2. Expected frequency in 400 drops: 0.62 × 400 = 248.
  3. Both are estimates. Another 150 drops would give a different count, and a second estimate from 1500 drops would be more reliable.

Your go. The first step is given. A basketball player makes 48 of her last 80 free throws. Estimate how many she will make in her next 50.

  1. The relative frequency of a made shot is 48 ÷ 80.
  2. Multiply by 50 to find the expected frequency.
Check the answer

48 ÷ 80 = 0.6, and 0.6 × 50 = 30 shots. It is an estimate: she might make 27 or 33. A common slip is 50 × 48 = 2400, which multiplies by the count and not by the proportion.

With a GDC (optional)

Most GDCs have a random-integer command, often called randInt( in a probability or number menu. Entering randInt(0, 1, 50) gives 50 random 0s and 1s, so counting the 1s simulates 50 coin flips. The order of the three numbers varies between models, so check the prompt. Press it again for a new run.

Store the results in a list, total them with the sum command and divide by the number of trials. A simulation gives an estimate that changes every time, which is the point of the lesson. Everything here can be done with a coin, a die or a table of random digits.

Check before you go on

1. Maya flips a coin 8 times and gets 6 heads. She says the coin must be biased. What is the best response?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A spinner is spun 240 times and lands on green 60 times. What is the relative frequency of green, as a decimal?

3. In Kuala Lumpur, the probability of rain on a given afternoon in November is 0.4. In how many of 60 afternoons would you expect rain?

Level 2 · Building

4. A factory tests 500 bulbs and finds 12 faulty. Using this, how many faulty bulbs would you expect in a batch of 8000?

5. A coin is flipped 50 times and gives 32 heads. It is then flipped 150 more times and gives 60 heads. Using all 200 flips, what is the experimental probability of heads, as a decimal?

6. A fair coin has come up tails 5 times in a row. What is the probability that the next flip is heads?

Level 3 · Stretch

7. A basketball player makes 70% of her shots. To simulate, random digits 0 to 6 mean a made shot and 7, 8 or 9 mean a miss. The digits 3, 8, 0, 6, 7 are generated for five shots. How many shots are made?

8. Each play of an arcade game wins a prize with probability 0.08. About how many plays are needed to expect 12 prizes?

Something to think about

Two people each flip a fair coin 1000 times. One says: “We will both get close to 500 heads, so we will get nearly the same number.” How close to 500 would you expect, and how far apart could the two totals be? Does the proportion of heads get closer to 0.5 even if the count does not?

One way to think about it

Totals of 480 and 520 would not be surprising: counts often differ from 500 by 15 or 20. The proportion is 0.48 and 0.52, so each is within 0.02 of 0.5. With 100 times as many flips the typical gap in the count grows (to about 150) while the proportion gets closer to 0.5 (within about 0.002). The proportion settles; the gap in the count does not shrink.

Exit ticket

9. A fair-looking die is rolled 40 times and shows a six 11 times. Which statement is best?