Each person in the group, one by one, announces his or her birthday. All members listen for a match, and as soon as one is found, the exercise stops. Surprisingly, a group of 30 or so members typically requires only six or seven announcements before a match is found. Include a graph of the row/column corresponding to 365 days.

Answer the following questions: 

(1) How many queries will you expect to make in a group of size 50 before finding a matching birthday? 

(2) What group size requires the most queries to find a birthday match, on average?

(3) Estimate the value that the expected number of queries approaches as the group size goes to infinity. Justify your estimate.