Example 6.19. Buffon tossed a coin 4040 times, with heads appearing 2048 times. Can the obtained deviation in the number of heads from 2020 be considered random, or is it due to a systematic cause?
Solution. The discrepancy between Buffon's empirical frequency and the theoretical one can be considered random if the probability that the deviation of the number of heads from 2020 is equal to or greater than Buffon's in absolute value when a coin is tossed 4040 times is sufficiently large. Let $m$ be the number of heads in 4040 coin tosses. We find the probability:

$$
P(|m-2020|<28)
$$

where $28=|2048-2020|$

$$
P(|m-n p|<\varepsilon) \approx \Phi \frac{\varepsilon}{\sqrt{n p q}}
$$

In this case, $p=0.5, q=0.5, n=4040, \varepsilon=28$. Therefore,

$$
\begin{gathered}
n p=4040 \cdot 0.5=2020 ; \quad \sqrt{n p q}=\sqrt{1010}=31.78 \\
n p q=2020 \cdot 0.5=1010 ; \quad \frac{\varepsilon}{\sqrt{n p q}}=\frac{28}{31.78}=0.881 \\
P(|m-2020|<28) \approx \Phi(0.881)=0.6217
\end{gathered}
$$

Therefore, the probability of the opposite event, i.e., that $|m-2020| \geq 28$, is $1-0.6217=0.3783$. Since this probability is sufficiently large, Buffon's result can be considered due to random causes.