
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A bag contains $4$ white balls and $2$ black balls. A ball is drawn from the bag each time.

$(1)$ If the ball is replaced after each draw, find the probability that the second draw is a black ball.

$(2)$ If the ball is not replaced after each draw, find the probability that the second draw is a black ball, given that the first draw is a white ball.

$(3)$ If the ball is replaced after each draw, find the distribution of the random variable $X$ representing the number of black balls drawn in three draws, and find $E(X)$.

Present the answer in LaTex format: \boxed{Your answer}
user