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Mathematics · Data and probability

Experiment and theory: experimental probability

If you toss a coin 10 times and get 7 heads, is the coin biased? Compare experimental and theoretical probability, see why results get closer to theory as the number of trials grows, and meet the gambler's fallacy.

  • 20 min
  • 1 simulation
  • Mini quiz

Updated:

Preview

The question

Deniz tossed a coin 10 times and it landed heads 7 times. "This coin is biased; the probability of heads isn't 12\dfrac{1}{2}!" says Deniz. Deniz's friend Burcu replies, "Toss it some more and let's see what happens." Who do you think is right? How do the results behave when we roll a dice or toss a coin many times?

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Number line

Experimental probability of a spinner

Drag the point or use the buttons to move in steps of 0.05. When you reach the right value, press Check.

Prediction

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If red came up 28 times in 80 spins, is the experimental probability less than or more than 0.5?

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Frequently asked

5 questions

If I got 7 heads in 10 tosses, is the coin unfair?

10 tosses are not enough to say that. With few tosses the share of heads can be quite far from 0.5. In the class experiment in the lesson the share was 0.7 after 10 tosses, 0.54 after 100 and 0.508 after 1000; it got closer to 0.5 as the number of tosses grew.

I got tails 5 times in a row. Is heads now more likely?

No, this way of thinking is called the gambler's fallacy. A coin has no memory; every toss is independent of the ones before. On the 6th toss the probability of heads is still 1/2.

If the coin doesn't make up for the past, why does the share get closer to the theoretical probability?

Because as the number of tosses grows, the first few tosses matter less and less in the total. Among 1000 tosses the first 5 make almost no difference; the coin does not "make up for" earlier tails, they simply lose their weight among many tosses.

Why do experiments if we have theoretical probability?

To work out a theoretical probability you need to be able to count equally likely outcomes. With a lopsided spinner, or the question of whether a drawing pin lands point up, the outcomes are not equally likely; there, experimental probability is the best tool for an estimate.

Will the expected number come out exactly?

No. n · P(A) is an approximate estimate. In 600 dice rolls you expect a 6 about 600 · 1/6 = 100 times, but it may not be exactly 100; you expect a number close to 100. That is also why it is normal that Ali got a 6 thirteen times in 60 rolls instead of 10.