Your brother Joe is a surgeon who suffers badly from the overconfidence bias. He loves to trade stocks and believes his predictions with $100\%$ confidence. In fact, he is uninformed like most investors. Rumours are that Vital Signs $($a start$-$up that makes warning labels in the medical industry$)$ will receive a takeover offer at $$20$ per share. Absent the takeover offer, the stock will trade at $$15$ per share. The uncertainty will be resolved in the next few hours. Your brother believes that the takeover will occur with certainty and has instructed his broker to buy the stock at any price less than $$20.$ In fact the true probability of a takeover is $50\%,$ but a few people are informed and know whether the takeover will actually occur. They also have submitted orders. Nobody else is trading in the stock.

a$.$ Describe what will happen to the market price if in fact the takeover occurs. What will your brother$$s profits be: positive, negative, or zero?

b$.$ Describe what will happen to the market if the takeover does not occur. What will your brother$$s profits be: positive, negative, or zero?

c$.$ What are your brother$$s expected profits?