Clare and noah play a game in which they earn the same number of points for each
goal and lose the same number of points for each penalty. clare makes 6 goals and 3
penalties, ending the game with 6 points. noah earns 8 goals and 9 penalties and ends
the game with −22 points.
a. write a system of equations that describes clare and noah's outcomes. use to
represent the number of points for a goal and to represent the number of
points for a penalty

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?