
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
18 scientists attended a conference, of which exactly 10 know a sensational piece of news. During the coffee break, the scientists randomly form pairs, and in each pair, anyone who knows the news shares it with the other if they don’t already know it.

a) Find the probability that after the coffee break, the number of scientists who know the news will be 13.

b) Find the probability that after the coffee break, the number of scientists who know the news will be 14.

c) Let $X$ be the number of scientists who know the sensational news after the coffee break. 

Find the expected value of $X$.

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