MathProbabilityProbabilistic modeling

Probabilities as frequencies

Proven or not proven?

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When we discussed the probability that a coin toss of 30 will land on heads at least 20 times, we ran the following code:

# imagine doing it yourself and not in python
NUMBER_OF_TRIALS = 100000
for i in range(10):
    occurrences = 0
    for j in range(NUMBER_OF_TRIALS):
        if at_least_20_heads(flip_coin_30_times()):
            occurrences += 1
    print(occurrences / NUMBER_OF_TRIALS, end=' ')

0.04935 0.04981 0.04911 0.048 0.04919 0.05057 0.0494 0.04827 0.04954 0.04883

Can we say that we have proven that this probability is close to 0.05 (say, from 0.04 to 0.06)?

Select one option from the list

No, because by conducting experiments we cannot prove anything in principle.

No, because we could have been very unlucky and in all 10 repetitions of 100000 trials, the proportion of trials in which the event occurred could have been far from the probability of the event.

Yes, according to the Borel's law of large numbers, all fractions tend towards the probability of an event. In our case, all fractions are close to 0.05, therefore the probability is close as well.

Yes, all fractions are between 0.04 and 0.06, so the probability is within the range.

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