Bellomy Research is a company in the U.S. who conducts research. Two types of companies were investigated: "large" and "small to mid-size".

The survey of 475 companies ("large" and "small to mid-sized") asked if they offer stock options to their board members, with only two possible answers "yes" or "no". All the respondents participated and answered the survey.

The results indicated that of 234 large companies 83 offered stock options. Of all small to mid-sized companies, 54 indicated that they offer stock options.

1. (2 point) Suppose that the respondent chosen is a small to mid-size company. What is the probability that it offered stock options?
2. (2 point) Suppose that the respondent chosen does indicate that it didn't offer stock options. What is the probability that the company is large?
3. (3 points) Are offering stock options and company type independent? Explain.
4. (1 point) What is the probability that a respondent chosen at random is a small to mid-sized company and indicates that it offers stock options?
5. (2 point) What is the probability that a respondent chosen at random is a small to mid-sized company or is a company who offers stock options?
6. (2 point) What is the probability that a respondent chosen at random is a large company or is a company who does not offer stock options?

Calculate all the above, show your work and present answer with 5 decimals