D2.2 · Compare theoretical and experimental probabilities for two events

Ontario Grade 7 Mathematics

Data

Ontario Grade 7 Mathematics — D2.2

What you will learn

• Explain the difference between theoretical and experimental probability.
• Find each probability using a fraction.
• Compare theoretical and experimental probabilities for two events and describe what the comparison shows.

Two ways to describe chance

• Theoretical probability uses wanted equally likely outcomes and all possible equally likely outcomes.
• Experimental probability uses the number of times an event happened and the total number of trials.
• Fractions help you compare the two kinds of probability.

Compare probabilities fairly

• Compare the same event in the theoretical and experimental results.
• Use the setup for theoretical probability and the trial record for experimental probability.
• State which probability is greater, smaller, or equal.

Organize and read the results

• Keep theoretical and experimental counts separate.
• Use the same probability scale for every bar.
• Make sure the chart agrees with the fractions.

A routine for comparing

• Write the probability as a fraction before comparing.
• Use matching denominators when they make the comparison clearer.
• Explain the comparison in words.

Keep the two kinds of probability separate

Worked example

A fair coin and two events

1. Find the theoretical probabilities
   A fair coin has two equally likely results. Heads is one result, and tails is the other. Each event is one of the two results.
   $$P_T(H)=\frac{1}{2},\quad P_T(T)=\frac{1}{2}$$
2. Find the experimental probabilities
   Use 10 as the total because there were 10 tosses. Heads happened 6 times and tails happened 4 times.
   $$P_E(H)=\frac{6}{10}=\frac{3}{5},\quad P_E(T)=\frac{4}{10}=\frac{2}{5}$$
3. Compare
   Three fifths is greater than one half, so heads had the greater experimental probability in these tosses. Two fifths is less than one half, so tails had the smaller experimental probability. The results do not change the theoretical probabilities for a fair coin.
   $$\frac{3}{5}>\frac{1}{2}>\frac{2}{5}$$

Worked example

Two colours of counters

1. Count the counters
   Add the green and yellow counters to find the total. Theoretical probability uses the counters in the bag.
   $$3+5=8$$
2. Find theoretical probabilities
   Three of the 8 counters are green, and 5 of the 8 are yellow. Each fraction describes one draw from this bag.
   $$P_T(G)=\frac{3}{8},\quad P_T(Y)=\frac{5}{8}$$
3. Find experimental probabilities
   Use the 16 trials as the total. Green happened 5 times and yellow happened 11 times.
   $$P_E(G)=\frac{5}{16},\quad P_E(Y)=\frac{11}{16}$$
4. Compare each event
   To compare green, rename three eighths as six sixteenths. Five sixteenths is less than six sixteenths. To compare yellow, rename five eighths as ten sixteenths. Eleven sixteenths is greater than ten sixteenths.
   $$\frac{5}{16}<\frac{3}{8},\quad \frac{11}{16}>\frac{5}{8}$$

Worked example

Compare two number-cube events

1. Count theoretical outcomes
   The even faces are 2, 4, and 6, so 3 of the 6 faces fit Event A. The faces greater than 4 are 5 and 6, so 2 of the 6 faces fit Event B.
   $$P_T(A)=\frac{3}{6}=\frac{1}{2},\quad P_T(B)=\frac{2}{6}=\frac{1}{3}$$
2. Use the trial results
   There were 18 rolls. Event A happened 8 times, and Event B happened 7 times. Use 18 as the denominator for each experimental probability.
   $$P_E(A)=\frac{8}{18}=\frac{4}{9},\quad P_E(B)=\frac{7}{18}$$
3. Compare within each event
   For Event A, one half is nine eighteenth parts and four ninths is eight eighteenth parts, so experimental probability is lower. For Event B, one third is six eighteenth parts, so seven eighteenth parts is greater.
   $$\frac{4}{9}<\frac{1}{2},\quad \frac{7}{18}>\frac{1}{3}$$

Common mistakes and how to avoid them

Lesson summary

• Theoretical probability comes from the possible equally likely outcomes in the setup.
• Experimental probability comes from recorded trials.
• Use wanted outcomes over all equally likely outcomes for theoretical probability.
• Use event occurrences over total trials for experimental probability.
• Compare the probabilities for each named event and explain what the comparison shows.

Check your understanding

Key terms

Event

An outcome, or group of outcomes, that we want to study.

Trial

One attempt, such as one toss, roll, or draw.

Theoretical probability

A probability found from the possible equally likely outcomes in a setup.

Experimental probability

A probability found from the results of trials that have happened.

Equally likely

Having the same chance of happening.

Continue through Grade 7 Math

View the complete Grade 7 Math Ontario Grade 7 Mathematics curriculum and lessons

• D1.6 · Critique circle graphs and misleading displays before drawing conclusions
• E1.1 · Classify cylinders, pyramids, and prisms by properties and symmetry
• D1.1 · Explain percent distributions of large populations and samples
• D1.2 · Collect and organize qualitative and quantitative data using percentages
• D1.3 · Select suitable graphs, including circle graphs
• D1.4 · Create a data infographic with tables and circle graphs

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation D2.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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