119. Find the probability that event $A$ will occur exactly 70 times in 243 trials, if the probability of this event occurring in each trial is 0.25.
Solution. According to the condition, $n=243 ; k=70 ; p=0.25 ; q=0.75$. Since $n=243$ is a sufficiently large number, we will use the local theorem of Laplace:

$$
P_{n}(k)=\frac{1}{\sqrt{n p q}} \varphi(x)
$$

where $x=(k-n p) / \sqrt{n p q}$.

Let's find the value of $x$:

$$
x=\frac{k-n p}{\sqrt{n p q}}=\frac{70-243 \cdot 0.25}{\sqrt{243 \cdot 0.25 \cdot 0.75}}=\frac{9.25}{6.75}=1.37
$$

From the table in Appendix 1, we find $\varphi(1.37)=0.1561$. The desired probability is

$$
P_{243}(70)=1 / 6.75 \cdot 0.1561=0.0231
$$