Example 6.15. Among the parts processed by a worker, on average $4 \%$ are non-standard. Find the probability that among 30 parts taken for testing, two parts will be non-standard. What is the most probable number of non-standard parts in the sample of 30 parts and what is its probability?
Solution. Here, the experiment involves checking each of the 30 parts for quality. Event $A$ - the appearance of a non-standard part; its probability $P=0.04$, then $q=0.96$. From this, using the Bernoulli formula, we find:

$$
P_{30}(2)=C_{30}^{2}(0.04)^{2}(0.96)^{28} \approx 0.202
$$

The most probable number of non-standard parts in this sample is calculated by formula (2):

$$
k_{0}=[30 \cdot 0.04+0.04]=[1.24]=1,
$$

and its probability is

$$
P_{30}(1)=C_{30}^{1} \cdot 0.04^{1} \cdot(0.96)^{29} \approx 0.305
$$