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Use Simulations to Estimate Compound Event Probabilities

Students design and use simulations (random number generators, spinners, dice, coin flips) to generate frequencies for compound events and estimate probabilities. They compare simulated results to theoretical probabilities and explain any differences.

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Use Simulations to Estimate Compound Event Probabilities

Grade 7 · Statistics and Probability · question 1 of 3

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20 rungs. A star at the top.

Emi rolled two number cubes 100 times, and in 13 of the rolls both cubes showed the same number. What is the estimate of the probability of doubles, as a decimal?

How it is taught

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A simulation must mimic the real chances.

A fair coin has two equally likely sides.

Read it: A simulation's estimate

Flip 2 coins together: that is one trial, and both heads is a hit. In 40 trials there were 9 hits, so the estimate of P(both heads) is 9 ÷ 40 = 0.225.

  1. Name one trial: the whole compound event, every part of it together.
  2. Name a hit: the outcome you are counting.
  3. Run many trials, count the hits, and divide: hits ÷ trials.

A compound event has more than one part, so one trial runs every part of it at once.

Each trial is a small experiment, and the more trials there are, the closer the estimate tends to come to the true chance.

The estimate is the hits' part of all the trials, a number from 0 to 1, like any probability.

An estimated probability is hits ÷ trials, where one trial is the whole compound event.

Counting each coin as its own trial: 9 ÷ 80.

One trial is both coins flipped together, so there are 40 trials: 9 ÷ 40 = 0.225.

Jo spun a spinner with 3 equal parts, one of them red, twice for each trial. She ran 200 trials, and both spins landed on red in 26 of them. What is the estimate of the probability of red twice, as a decimal? — 0.13. One trial is 2 spins; a hit is red both times. 26 hits in 200 trials: 26 ÷ 200 = 0.13.

Estimating P(two heads) with 20 simulated coin flips

We're flipping two coins together. What's the chance both land on heads?

First, the theory. Two coins have four equally likely outcomes: HH, HT, TH, and TT.

Only one of those four is HH, so the theoretical probability is one fourth, or 0.25.

But what if we didn't know the theory? We could run a simulation. Let's flip two coins twenty times.

Here are the results of all twenty trials.

Now we count the trials that landed HH. There are five of them.

Five out of twenty trials. That's our experimental estimate of the probability.

Five over twenty equals 0.25 — the same as the theoretical chance. With more trials, our estimate would settle even closer.

Simulations let us estimate the probability of compound events by counting how often they happen.

Two coins — what is P(HH)?(both coins land heads)Sample space — all 4 outcomesHHHTTHTT1 of 4 outcomes is HH14Simulation — 20 two-coin flipsHTTTHHTHHTHHTTHTTHHHHTTTHHTHHTTTHTHHTHTT5 trials are HHEstimate: 5 ÷ 20 = 0.25Theory 0.25 ≈ Estimate 0.25P(HH) ESTIMATE0.255 of 20 trials
Grade 7 · Simulating two coin flips to estimate P(HH)

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
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Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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