Exercise $1$:

A survey by the Arthur Andersen Enterprise attempted to determine what the leading challenges are for the growth and survival of small businesses. Although the economy and finding qualified workers were the leading challenges, several others were listed in the results of the study, including regulations, listed by $30\%$ of the companies, and the tax burden, listed by $35\%.$ Suppose that $71\%$ of the companies listing regulations as a challenge also listed the tax burden as a challenge. Assume these percentages hold for all small businesses.

If a small business is randomly selected, find the following probabilities:

$($a$)$ The small business lists either the tax burden or regulations as a challenge.

$($b$)$ The small business lists either the tax burden or regulations but not both as a challenge.

$($c$)$ The small business does not list regulations as a challenge given that it lists the tax burden as a challenge.

$($d$)$ The small business does not list regulations as a challenge given that it does not list the tax burden as a challenge.