42. The probability of the establishment of a stable snow cover in a certain area from October is 0.2. What is the probability that in the next two years in this area a stable snow cover from October

a) will not establish once;

b) will establish at least once?
Solution. Let's denote the events:

$A_{1}$ - a stable snow cover from October will establish in the first year;

$A_{2}$ - a stable snow cover from October will establish in the second year;

$\bar{A}_{1}$ - a stable snow cover from October will not establish in the first year;

$\bar{A}_{2}$ - a stable snow cover from October will not establish in the second year;

$B$ - a stable snow cover from October will not establish in the next two years;

$A$ - a stable snow cover from October will establish at least once in the next two years.

The probabilities of events $A_{1}$ and $A_{2}$ are: $P\left(A_{1}\right)=P\left(A_{2}\right)=p=0.2$. The probabilities of the opposite events $\bar{A}_{1}$ and $\bar{A}_{2}$ are:

$$
P\left(\bar{A}_{1}\right)=P\left(\bar{A}_{2}\right)=1-p=q=0.8.
$$

a) Event $B$ consists in the simultaneous occurrence of events $\bar{A}_{1}$ and $\bar{A}_{2}$, therefore $B=\bar{A}_{1} \bar{A}_{2}$. Events $\bar{A}_{1}$ and $\bar{A}_{2}$ are independent; the probability of event $\boldsymbol{B}$ will be found using the multiplication theorem of probabilities of independent events:

$$
P(B)=P\left(\bar{A}_{1}\right) \cdot P\left(\bar{A}_{2}\right)=q^{2}=0.64
$$

b) The probability of event $A$ will be found using formula (**) from problem 40 for $n=2:$

$$
P(A)=1-q^{2}=0.36
$$