Optometry school applicants decreased by 25% during a 4-year period. during the same time, the number of first-year openings in optometry school increased by 12%. if the ratio of applicants to first-year student openings had been 3 to 1, then which of the following would be the approximate ratio at the end of the 4-year period? a. 1.5 to 1b. 2 to 1c. 3 to 2d. 4 to 3e. 6 to 5e. ((63.5)/(7.)

You are in a room of 22 people (including yourself). each of you has their own deck of 52 cards — assume each deck is well-shuffled, and can be assumed to be fair and independent of each other.1. each you draws the top card of your deck.what is the probability that at least one other person has drawn the same card as you? 2. let x be the number of people who drew a card that was also drawn by at least one other person. for example, if you all drew the same card, x=22, but if you all drew different cards, x=0. what is the expected value of x? 3. consider the same situation as the last question, but this time with 29 people: all 29 of you are in a room with your own decks of cards, and you draw one of them.what is the probability that at least one person draws the ace of spades? 4. let x be the number of the types of card drawn. for example, if everyone draws the same card, x=1, but if everyone draws a different card, x=29. what is the expected value of x?