Experimental probability and expected outcomes

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课程笔记

What is experimental probability?

  • Experimental probability is an estimate of how likely an event is, based on actually carrying out an experiment.
  • It is found by doing trials and recording what happens, not by assuming a theoretical model.
  • Another name for it is relative frequency.
  • It can also be called empirical probability or a-posteriori probability.
  • It is based on observation and experience rather than deductive reasoning.

Calculating experimental probability

  • For an event A, the experimental probability is P(A) = \frac{m}{n}, where m = number of trials in which A happens and n = total number of trials.
  • The numerator m counts only the outcomes where the event occurs.
  • The denominator n is the total number of trials carried out.
  • The result is a fraction, decimal or percentage between 0 and 1.
  • Example: if a drawing pin lands point-up 30 times in 50 drops, the experimental probability is 30/50 = 0.6.

More trials give a better estimate

  • A small number of trials can give a very unreliable estimate.
  • As the number of trials increases, the experimental probability usually gets closer to the true probability.
  • This happens because random variation has less effect when the sample is larger.
  • So a result from 1000 trials is generally more trustworthy than one from 10 trials.
  • Very large numbers of trials are needed to estimate probabilities that are close to 0 or close to 1 accurately.

Experimental vs theoretical probability

  • Theoretical probability is worked out from a sample space, assuming all outcomes are equally likely.
  • Experimental probability is worked out from the results of an actual experiment.
  • They are often different, especially when only a few trials are done.
  • With many trials, the experimental probability tends to get closer to the theoretical probability.
  • Comparing the two helps you decide whether a dice, spinner or coin is fair or biased.

Expected outcomes

  • The expected number of times an event happens is: expected number = probability × number of trials.
  • Use the probability of the event (theoretical or experimental) and multiply by how many trials you plan to do.
  • Example: if the probability of scoring a 6 is 1/6 and you roll the dice 60 times, the expected number of 6s is 1/6 × 60 = 10.
  • Expected outcomes are predictions, not guarantees — the actual result may differ.
  • The more trials you do, the closer the actual results usually get to the expected number.

Fairness and bias

  • A fair dice or coin has equal theoretical probability for each outcome.
  • If experimental probabilities are very different from the theoretical ones after many trials, the object may be biased.
  • A few unusual results do not prove bias — you need enough trials to be confident.
  • Example: 100 spins of a spinner give 40 reds; if red should be 1/4, this suggests possible bias.
  • Judging fairness means comparing experimental results with what theory predicts.

Advantages and disadvantages

  • Advantage: experimental probability needs few assumptions — you just count what happens.
  • It can estimate probabilities for events where no theoretical model is available.
  • Disadvantage: it can be inaccurate when the number of trials is small.
  • It is poor at estimating probabilities very close to 0 or very close to 1 unless the sample is huge.
  • A statistical model can sometimes give a better estimate, but only if its assumptions are true.

Key vocabulary

  • Trial — one repeat of the experiment (one roll, one spin, one drop).
  • Event — the outcome or set of outcomes you are interested in.
  • Relative frequency — another name for experimental probability.
  • Sample space — the set of all possible outcomes.
  • A priori probability — a probability based on reasoning, not on observations.

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练习题

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  1. 1.Which formula gives the experimental probability of an event?

    Easy
    • Anumber of trials ÷ number of times the event occurs
    • Bnumber of times the event occurs ÷ number of trials
    • Cnumber of times the event occurs × number of trials
    • Dnumber of trials − number of times the event occurs
  2. 2.A fair coin is tossed 50 times and lands on heads 23 times. What is the experimental probability of heads?

    Easy
    • A0.23
    • B0.46
    • C0.50
    • D0.54
  3. 3.A spinner is spun 200 times and lands on red 64 times. Which of these is the experimental probability of red?

    Easy
    • A0.32
    • B0.64
    • C0.68
    • D0.36
  4. 4.A bag contains 5 red and 3 blue counters. A counter is taken and replaced 40 times, and a red counter is taken 26 times. Which statement is true?

    Easy
    • AThe experimental probability of red is 0.65, which is greater than the theoretical probability of 0.625.
    • BThe experimental probability of red is 0.65, which is less than the theoretical probability of 0.625.
    • CThe experimental probability of red is 0.625, which equals the theoretical probability.
    • DThe experimental probability of red is 0.35, which is less than the theoretical probability of 0.625.
  5. 5.In general, the more trials you do in an experiment, the better the experimental probability estimates the true probability.

    Easy

    True or false?

  6. 6.Select all the statements that are true about experimental probability. (Select all that apply.)

    Medium
    • AIt is also called relative frequency.
    • BIt is calculated as the number of successful outcomes divided by the total number of trials.
    • CIt always equals the theoretical probability.
    • DIt can be estimated from observation and experience.
    • EIt requires a theoretical sample space to calculate.
  7. 7.A fair dice is rolled 90 times. How many times would you expect to roll a 4?

    Medium
    • A10
    • B15
    • C20
    • D30
  8. 8.A biased coin has probability 0.3 of landing on heads. If the coin is tossed 200 times, how many heads would you expect?

    Medium
    • A30
    • B60
    • C70
    • D140

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