Practice Maths

L18 — Comparing Probabilities

Theoretical vs Experimental Probability

Theoretical Probability is what we expect to happen based on maths.
Experimental Probability (also called relative frequency) is what actually happens when we carry out an experiment.

Key Terms

theoretical probability
the expected probability, calculated from the number of favourable outcomes divided by the total number of possible outcomes
experimental probability
probability calculated from actual results of trials; also called relative frequency
relative frequency
the number of times an event occurred divided by the total number of trials (e.g. 8 sixes in 30 rolls gives relative frequency = 830)
trial
one single performance of an experiment (e.g. one coin flip, one die roll)
equally likely
when all outcomes have the same probability of occurring (e.g. each face of a fair die)
Relative Frequency Formula
Experimental probability = frequency of the eventtotal number of trials

Worked Example

Question: Compare P(rolling a 5 on a die) with P(flipping Heads on a coin).

Step 1 — Calculate first probability: P(5 on die) = 16

Step 2 — Calculate second probability: P(Heads) = 12 = 36

Step 3 — Compare: Since 36 > 16, flipping Heads is more likely than rolling a 5.

Probability as a Number

Probability measures how likely an event is to happen. We express it as a number between 0 and 1:

  • Probability of 0 means the event is impossible (e.g. rolling a 7 on a standard die).
  • Probability of 1 means the event is certain (e.g. getting a number less than 7 when rolling a standard die).
  • Probability of 12 means the event is equally likely to happen or not (e.g. getting Heads on a fair coin).

The closer the probability is to 1, the more likely the event. The closer it is to 0, the less likely.

The Probability Formula

For a fair experiment (equally likely outcomes):

P(event) = number of favourable outcomes ÷ total number of outcomes

  • Rolling a 3 on a die: 1 favourable out of 6 total → P(3) = 16
  • Drawing a red card from a standard deck of 52: 26 red cards → P(red) = 2652 = 12
  • Getting a vowel from the letters A, B, C, D, E: 2 vowels (A, E) out of 5 → P(vowel) = 25

Placing Events on a Number Line

You can represent and compare probabilities by placing them on a number line from 0 to 1. Events further to the right are more likely. For example:

  • P(rolling an even number on a die) = 36 = 12 → sits at the midpoint
  • P(rolling a number greater than 1) = 56 → sits close to 1 (very likely)
  • P(rolling a 6) = 16 → sits close to 0 (not very likely)

Comparing Probabilities

To compare two probabilities, convert them to the same denominator (or to decimals) and compare. For example:

  • P(A) = 38 and P(B) = 25. Convert: 38 = 1540, 25 = 1640. So P(B) > P(A) — event B is more likely.

When comparing multiple events, list them from least likely (closest to 0) to most likely (closest to 1).

Key tip: Probability is always between 0 and 1 inclusive. If you calculate a probability greater than 1 or less than 0, you have made an error. Also remember: a probability of 14 does NOT mean the event will happen exactly once every 4 tries — it means in the long run, it happens about 1 in every 4 times on average.

Practice Questions

  1. Theoretical and Experimental Probability Fluency

    1. What is “theoretical probability”? Write a definition in one sentence.
    2. A student flips a coin 20 times and gets Heads 9 times. What is the experimental probability of Heads? Write as a fraction.
    3. What is the theoretical probability of Heads on a fair coin? Write as a fraction.
    4. Are your answers in (b) and (c) the same? If not, does this mean the coin is unfair? Explain.
    5. If the student flipped the coin 500 times instead of 20, would you expect the experimental probability to be closer to or further from the theoretical probability? Explain.

  2. Likelihood Descriptions Fluency

    Describe each event using one of these terms: impossible, unlikely, equally likely, likely, certain.

    1. Rolling a 10 on a standard 6-sided die.
    2. Tossing a coin and it landing on Heads.
    3. Drawing a red card from a standard deck (26 red cards out of 52).
    4. Rolling a number less than 7 on a standard 6-sided die.
    5. Randomly drawing a blue marble from a bag containing 9 red marbles and 1 blue marble.

  3. Theoretical Probability — The Lucky Dip Fluency

    A bucket contains 10 green balls, 5 yellow balls, and 5 red balls (20 in total). A ball is drawn at random.

    1. Find P(green). Write as a fraction in simplest form and as a decimal.
    2. Find P(red). Write as a fraction in simplest form and as a decimal.
    3. Find P(yellow). Write as a fraction in simplest form and as a decimal.
    4. What is P(yellow or red)?
    5. What is P(not green)?

  4. Comparing Probabilities Fluency

    1. Which is more likely: P(A) = 13 or P(B) = 25? Convert to a common denominator to decide.
    2. Which is greater: 0.6 or 35? Show working.
    3. Order from least to most likely: 34, 0.6, 70%.
    4. Order from least to most likely: 13, 0.4, 45%.
    5. A student says: “14 > 13 because 4 > 3.” Explain the error and state the correct comparison.

  5. Experimental Probability — The Tally Understanding

    A student draws a counter from a bag 40 times, replacing it each time. Here are the results:

    ColourFrequency
    Blue18
    Orange22
    1. Calculate the experimental probability of drawing Blue. Write as a decimal.
    2. Calculate the experimental probability of drawing Orange. Write as a decimal.
    3. If the bag contains an equal number of Blue and Orange counters, what is the theoretical probability of Blue?
    4. Does the experimental result prove the bag is NOT equally split? Explain.

  6. Probability Number Line Understanding

    Use the probability number line below as a reference for this question.

    0 0.25 0.5 0.75 1 Impossible Equally Likely Certain
    1. Where on the number line would P = 16 sit (as a decimal)? Describe its likelihood.
    2. Where would P = 56 sit? Describe its likelihood.
    3. Convert each of these to decimals and order them from least to most likely on the number line: P(rolling a prime on a die) = 36, P(drawing a heart from a deck) = 1352, P(tossing Heads) = 12. What do you notice?
    4. Order these from least to most likely: 0.2, 13, 55%, 34.

  7. Dice Challenge Understanding

    1. List the sample space for rolling a standard 6-sided die.
    2. What is the theoretical probability of rolling a prime number (2, 3 or 5)? Write as a fraction in simplest form.
    3. In 60 rolls, a student rolled a prime number 28 times. Calculate the experimental probability of rolling a prime. Write as a fraction.
    4. Is the experimental probability from (c) above or below the theoretical probability? Show your comparison with working.

  8. Are They Equally Likely? Understanding

    1. A bag contains 4 red balls and 6 blue balls. Is the probability of picking red equal to the probability of picking blue? Show working and explain.
    2. A spinner has 4 sections but two sections are twice as large as the other two. Are all outcomes equally likely? Explain.
    3. A bag contains 5 red and 5 blue balls. Are P(red) and P(blue) equal? If 100 balls are drawn (with replacement), how many red would you theoretically expect?
    4. Give one example of an experiment where all outcomes are equally likely, and one where they are not.

  9. Working Backwards from Relative Frequency Problem Solving

    In a game, the relative frequency of winning was 0.2 after 50 trials.

    1. How many times did the player win in those 50 trials?
    2. With 500 trials instead of 50, would you expect the relative frequency to be closer to or further from the true probability? Explain why.
    3. After 500 total trials, the player had won a total of 95 times. Calculate the relative frequency after 500 trials and compare to the initial result of 0.2.

  10. The Fête Games Problem Solving

    Three games are set up at a school fête:

    • Game A: Roll a standard die; win if you roll a 6.
    • Game B: Spin a 10-section spinner; win if you land on one of 3 sections.
    • Game C: Draw a card from a standard deck of 52; win if you draw a heart.
    1. Calculate the probability of winning each game. Write each as a fraction.
    2. Rank the three games from least likely to most likely to win.
    3. If you play each game 60 times, how many wins would you theoretically expect from each? Show working for all three games.
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