68. For the red clover variety Perm local, on average $84\%$ of the plants are late maturing. What is the probability that 52 plants out of 60 clover plants, randomly selected, will be late maturing?
S o l u t i o n. In each trial, it is determined whether the clover plant is late-maturing. The total number of independent trials is $n=60$. Event $A$, which may or may not occur in each trial, consists in the fact that the selected plant is late-maturing. The probability of event $A$ occurring in each trial is $p=0.84$; the probability of event $A$ not occurring in each trial is $q=0.16$; the number of occurrences of event $A$ is $m=52$. We need to find $P_{60}(52)$.

The number $n$ of trials is large, so to find $P_{60}(52)$, we will use the asymptotic Laplace formula (20), which allows us to calculate the required probability approximately but with a high degree of accuracy.

Let's determine

$$
x=\frac{m-n p}{\sqrt{n p q}}=\frac{52-60 \cdot 0.84}{\sqrt{60 \cdot 0.84 \cdot 0.16}} \approx 0.5634
$$

From the table of values of the function $\varphi(x)$ (see table 1 in the appendix), we find $\varphi(0.56)=0.3410$.

The required probability is approximately equal to

$$
P_{60}(52) \approx \frac{\varphi(x)}{\sqrt{n p q}}=\frac{0.3410}{\sqrt{60 \cdot 0.84 \cdot 0.16}} \approx 0.1201 .
$$