4. (12 points) The board of directors of Tar Heel Industries consists of eight men and four women. A three-member search committee is to be chosen at random from among the board of directors to conduct a nationwide search for a new company president. (Note that because the search committee members are chosen "without replacement\" from the board, the selections are not independent trials.) 4.1. If the rst person chosen is a woman, what is the probability that the second person chosen is also a woman? 4.2. If the first two people chosen are women, what is the probability that the third person chosen is also a woman? 4.3. What is the probability all three members of the search committee will be women? 5. * (10 points) Email spam unsolicited messages sent in bulk is a nuisance for email users and providers. One estimate indicates that 45 percent of all email sent in 2021 was spam. Suppose that Alphabet, the parent company of Google, has hired you to improve the spam detection in Gmail, an email service. You use recent advances in machine learning to build a system that will correctly identify 95 percent of spam messages, while only incorrectly agging 1 percent of valid messages as spam. If your system identies a message as spam, what is the probability that it is actually spam