A man might go to cut trees tomorrow according to the following conditions:

$$ If the weather is clear in the morning, there is an $80\%$ chance that he will go to cut trees.

$$ If the weather is cloudy in the morning, there is a $40\%$ chance that he will go to cut trees.

$$ If the weather is rainy in the morning, there is a $10\%$ chance that he will go to cut trees.

$$ Tomorrow, there is a $30\%$ chance of clear weather, a $50\%$ chance of cloudy weather, and a $20\%$ chance of rainy weather.

Answer the following questions.

$($a$)$ What is the probability that Nasreddin Hodja will go to cut trees tomorrow?

$($b$)$ If the man did not go to cut trees for two consecutive days, what is the probability that the weather was clear on both of these days?