experimental probability

(noun)

The probability that a certain outcome will occur, as determined through experiment.

Related Terms

Examples of experimental probability in the following topics:

  • Experimental Probabilities

    • The experimental probability is the ratio of the number of outcomes in which an event occurs to the total number of trials in an experiment.
    • The experimental (or empirical) probability pertains to data taken from a number of trials.
    • $\displaystyle \text{experimental probability of event} = \frac{\text{occurrences of event}}{\text{total number of trials}}$
    • Experimental probability contrasts theoretical probability, which is what we would expect to happen.
    • If we conduct a greater number of trials, it often happens that the experimental probability becomes closer to the theoretical probability.

  • Theoretical Probability

    • Probability theory uses logic and mathematical reasoning, rather than experimental data, to determine probable outcomes.
    • Mathematically, probability theory formulates incomplete knowledge pertaining to the likelihood of an event.
    • This probability is determined through measurements and logic, but not through any experimental findings (the future has not yet happened).
    • For example, the probability of rolling any specific number on a six-sided die is one out of six: there are six, equally probable sides to land on, and each side is distinct from the others.
    • This is a theoretical probability; testing by rolling the die many times and recording the results would result in an experimental probability.

  • Example Calculations

    • However, the experimenter does not know and asks Mr.
    • Therefore, the experimenter will determine how many times Mr.
    • The question is: what is the probability the experimenter will correctly reject the null hypothesis that π = 0.50?
    • The question, then, is what is the probability the experimenter gets a sample mean greater than 78.29 given that the population mean is 80?
    • Therefore, the probability that the experimenter will reject the null hypothesis that the population mean for the new method is 75 or lower is 0.80.

  • Randomization Tests: Two Conditions

    • The data in Table 1 are from a fictitious experiment comparing an experimental group with a control group.
    • The scores in the Experimental Group are generally higher than those in the Control Group with the Experimental Group mean of 14 being considerably higher than the Control Group mean of 4.
    • This means that if assignments to groups were made randomly, the probability of this large or a larger advantage of the Experimental Group is 3/70 = 0.0429.
    • Since only one direction of difference is considered (Experimental larger than Control), this is a one-tailed probability.
    • The two-tailed probability is 0.0857 since there are 6/70 ways to arrange the data so that the absolute value of the difference between groups is as large or larger than the one obtained.

  • Randomization Tests: Contingency Tables: (Fisher's Exact Test

    • Three of the 4 subjects in the Experimental Group and none of the subjects in the Control Group solved the problem.
    • For this example, the probability is
    • This is a one-tailed probability since it only considers outcomes as extreme or more extreme favoring the Experimental Group.
    • Therefore, the two-tailed probability is 0.1428.
    • Note that in the Fisher Exact Test, the two-tailed probability is not necessarily double the one-tailed probability.

  • Lab: Probability Topics

    • The student will use theoretical and empirical methods to estimate probabilities.
    • Use the information from this table to complete the theoretical probability questions.
    • Use the data from the "Empirical Results" table to calculate the empirical probability questions.
    • Would this change (see (3) above) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart?
    • Hint: Think about the sample space for each probability.

  • Interpreting Non-Significant Results

    • Let's say Experimenter Jones (who did not know π = 0.51) tested Mr.
    • However, we know (but Experimenter Jones does not) that π = 0.51 and not 0.50 and therefore that the null hypothesis is false.
    • The experimenter should report that there is no credible evidence Mr.
    • In other words, the probability value is 0.11.
    • Using a method for combining probabilities, it can be determined that combining the probability values of 0.11 and 0.07 results in a probability value of 0.045.

  • Categorical Data and the Multinomial Experiment

    • In probability theory, the multinomial distribution is a generalization of the binomial distribution.
    • For $n$ independent trials, each of which leads to a success for exactly one of $k$ categories and with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.
    • The binomial distribution is the probability distribution of the number of successes for one of just two categories in $n$ independent Bernoulli trials, with the same probability of success on each trial.
    • In a multinomial distribution, the analog of the Bernoulli distribution is the categorical distribution, where each trial results in exactly one of some fixed finite number $k$ of possible outcomes, with probabilities $p_1, \cdots , p_k$ (so that $p_i \geq 0$ for $i = 1, \cdots, k$ and the sum is $1$), and there are $n$ independent trials.
    • The probabilities of the $k$ outcomes, denoted by $p_1$, $p_2$, $\cdots$, $p_k$, remain the same from trial to trial, and they sum to one.

  • Exercises

    • The probability value is the probability of obtaining a statistic as different (add three words here) from the parameter specified in the null hypothesis as the statistic obtained in the experiment.
    • The probability value is computed assuming that the null hypothesis is true.

  • Causation

    • Consider a simple experiment in which subjects are sampled randomly from a population and then assigned randomly to either the experimental group or the control group.
    • The fact that the greater stress in the control group was due to chance does not mean it could not be responsible for the difference between the control and the experimental groups.
    • In other words, the observed difference in "minutes slept" could have been due to a chance difference between the control group and the experimental group rather than due to the drug's effect.
    • If that probability is low, then it is inferred (that's why they call it inferential statistics) that the treatment had an effect and that the differences are not entirely due to chance.
    • Of course, there is always some nonzero probability that the difference occurred by chance so total certainty is not a possibility.