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Probability: experimental probability — KCSE Mathematics

KCSE Mathematics · 108 practice questions · 3 syllabus objectives · 3 revision lessons

36 easy36 medium36 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Perform simple experiments (tossing a coin, rolling a die) to collect relative frequency data; state the relationship between relative frequency and theoretical probability

Explain that as the number of trials increases, experimental probability approaches theoretical probability (law of large numbers)

Probability: experimental probability

Revision Notes

Concise lesson notes for Probability: experimental probability, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Experimental Probability

In probability, experimental probability is determined by conducting experiments and recording outcomes. It is calculated as:

  • Experimental Probability (P) = Number of favorable outcomes / Total number of trials

For example, if you toss a coin 100 times and it lands on heads 55 times, the experimental probability of getting heads is:

  • P(Heads) = 55 / 100 = 0.55

The theoretical probability is the expected probability based on possible outcomes. For a fair coin, the theoretical probability of heads is:

  • P(Heads) = 1/2 = 0.5

Relationship: As the number of trials increases, the experimental probability tends to approach the theoretical probability. This is known as the Law of Large Numbers. Thus, if we conduct more tosses, our experimental results will align closer to the theoretical values.

To summarize:

  • Experimental probability is based on actual experiments.
  • Theoretical probability is based on expected outcomes.
  • With more trials, experimental probability approximates theoretical probability.

Key points to remember

  • Experimental probability is calculated from actual experiments.
  • Theoretical probability is based on possible outcomes.
  • More trials lead to a closer approximation of theoretical probability.
  • Law of Large Numbers states this convergence over many trials.

Worked example

Question: If you roll a die 60 times and get a 4, 10 times, what is the experimental probability of rolling a 4? Answer: P(4) = 10 / 60 = 1/6. The theoretical probability is also 1/6.

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Lesson 2: Understanding Experimental Probability

Objective: Explain that as the number of trials increases, experimental probability approaches theoretical probability (law of large numbers)

Experimental probability is defined as the ratio of the number of successful outcomes to the total number of trials conducted. According to the law of large numbers, as the number of trials increases, the experimental probability will tend to approach the theoretical probability. This means that with more trials, our results become more reliable and closer to the expected outcome.

For example, if we toss a fair coin 10 times and get heads 6 times, the experimental probability of getting heads is 6/10 or 0.6. However, if we increase the number of tosses to 1000 and find heads 510 times, the experimental probability becomes 510/1000 or 0.51. This value is much closer to the theoretical probability of 0.5.

Key points to remember:

  • Experimental probability is based on actual trials.
  • Theoretical probability is the expected outcome based on mathematical reasoning.
  • As trials increase, experimental results converge to theoretical predictions.
  • This convergence illustrates the law of large numbers.
  • Experimental probability is based on actual experiments.
  • Theoretical probability is the expected outcome.
  • Increasing trials leads to closer results to theoretical probability.
  • The law of large numbers explains this convergence.

A die is rolled 60 times, resulting in a 4 appearing 15 times.

  • Experimental probability of rolling a 4: 15/60 = 0.25.
  • Theoretical probability of rolling a 4: 1/6 ≈ 0.167.
  • As trials increase, experimental probability approaches theoretical probability.
Lesson 3: Understanding Experimental Probability

Objective: Probability: experimental probability

Experimental probability is determined by conducting an experiment and recording the outcomes. It is calculated using the formula:

Experimental Probability (P) = Number of favorable outcomes / Total number of trials

To find the experimental probability, follow these steps:

  1. Conduct the experiment multiple times.
  2. Count the number of times the event of interest occurs.
  3. Divide by the total number of trials.

For example, if you toss a coin 100 times and it lands on heads 45 times, the experimental probability of getting heads is:

P(heads) = 45 / 100 = 0.45.

This means there is a 45% chance of getting heads based on your experiment. Remember, experimental probability can vary with different trials, but it gives a practical insight into the likelihood of events based on actual data.

  • Experimental probability is based on actual experiments.
  • Use the formula: P = favorable outcomes / total trials.
  • Results may vary with different trials.
  • More trials lead to more accurate probabilities.

If a die is rolled 60 times, landing on 4 a total of 12 times, find the experimental probability of rolling a 4.

P(4) = 12 / 60 = 0.2, or 20%.

Sample Questions

Read 3 questions and answers free. Sign up to access all 108 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

In a game, a player rolls a fair die 120 times and records the results: 1 - 20 times, 2 - 25 times, 3 - 30 times, 4 - 15 times, 5 - 20 times, 6 - 10 times. (a) Calculate the experimental probability of rolling a 3. [2 marks] (b) If the player rolls the die 300 more times, estimate the number of times a 3 is expected to appear. [2 marks]

Answer & marking scheme

Part (a) — 2 marks
P(3) = 30/120 or 0.25 stated correctly (1 mk)
Experimental probability of rolling a 3 = 0.25 (1 mk)
Part (b) — 2 marks
P(3) = 0.25 used correctly (1 mk)
Estimated number for next rolls = 0.25 × 300 = 75 stated correctly (1 mk)
2
easySHORT ANSWER4 marks

During an experiment, a student flips a coin 80 times and records 45 heads and 35 tails. (a) Calculate the experimental probability of getting heads. [2 marks] (b) If the student flips the coin 40 more times, how many heads would you expect to get? [2 marks]

Answer & marking scheme

Part (a) — 2 marks
P(heads) = 45/80 simplified to 9/16 or 0.5625 (1 mk)
State the probability correctly as a fraction or decimal (1 mk)
Part (b) — 2 marks
P(heads) = 45/80 calculated as 0.5625 (1 mk)
Expected heads in 40 rolls = 0.5625 × 40 = 22.5 or 22 or 23 depending on rounding (1 mk)
3
easySHORT ANSWER4 marks

A student rolls a six-sided die 60 times and records the results as follows: 10 ones, 15 twos, 12 threes, 8 fours, 9 fives, and 6 sixes. (a) Calculate the experimental probability of rolling a three. [2 marks] (b) How many times would you expect to roll a five if the die is rolled 100 more times? [2 marks]

Answer & marking scheme

Part (a) — 2 marks
P(three) = 12/60 simplified to 1/5 or 0.2 (1 mk)
State the probability correctly as a fraction or decimal (1 mk)
Part (b) — 2 marks
P(five) = 9/60 calculated as 0.15 (1 mk)
Expected fives in 100 rolls = 0.15 × 100 = 15 (1 mk)
4

A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is drawn at random 40 times, resulting in 16 red, 12 blue, and 12 green balls. (a) State the experimental probability of drawing a red ball. (b) If another 80 draws are made, estimate the number of blue balls expected to be drawn. (3 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Probability: experimental probability" cover?

Probability: experimental probability covers Perform simple experiments (tossing a coin, rolling a die) to collect relative frequency data; state the relationship between relative frequency and theoretical probability; Explain that as the number of trials increases, experimental probability approaches theoretical probability (law of large numbers); Probability: experimental probability, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Probability: experimental probability?

HighMarks has 108 Probability: experimental probability practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Probability: experimental probability for the KCSE exam?

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