A departmental store sends bills to charge its customers once a month. Past experience shows that 70 % of its customers pay their first month bill in…

A departmental store sends bills to charge its customers once a month. Past experience shows that $\displaystyle 70 \%$ of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of $\displaystyle 0.8$ of paying in time next month and the customer who doesn't pay in time has the probability of $\displaystyle 0.4$ of paying in time the next month. Based on the above information, answer the following questions:

Let $\displaystyle \mathrm{E}_{1}$ and $\displaystyle \mathrm{E}_{2}$ respectively denote the event of customer paying or not paying the first month bill in time. Find $\displaystyle \mathrm{P}\left(\mathrm{E}_{1}\right), \mathrm{P}\left(\mathrm{E}_{2}\right)$.

Let A denotes the event of customer paying second month's bill in time, then find $\displaystyle \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)$ and $\displaystyle \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)$.

Find the probability of customer paying second month's bill in time.

Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.