Random variables 2.3 The Numbers game Let's play a game! I have two envelopes. Each contains an integer in the range 0, 1,...,100 and the numbers are distinct. To win the game, you must determine which envelope con tains the larger number. To give you a fighting chance, I'll let you peek at the number in one envelope selected at random. Can you devise a strategy that gives you a better than 50% chance of winning? For example, you could just pick an evelope at random and guess that it contains the larger number. But this strategy wins only 50% of the time. Your challenge is to do better So you might try to be more clever. Suppose you peek in the left envelope and see the number 12. Since 12 is a small number, you might guess thatthat other number is larger. But perhaps I'm sort of tricky and put small numbers in both envelopes. Then your guess might not be so good! An important point here is that the numbers in the envelopes may not be random I'm picking the numbers and I'm choosing them in a way that I think will defeat your guessing strategy. I'll only use randomization to choose the numbers if that serves my end: making you lose 2.3.1 Intuition Behind the Winning Strategy Amazingly, there is a strategy that wins more than 50% of the time, regardless of what numbers i put in the envel Suppose that you somehow knew a number a between my lower number and higher numbers. Now you peek in an envelope and see one or the other. If it is bigger than r, then you know you' re peeking at the higher number. If it is smaller than then you're peeking at the lower number. In other words, if you know an number z between my lower and higher numbers, then you are certain to win the game The only flaw with this brilliant strategy is that you do not know c. Oh well But what if you try to guess a? There is some probability that you guess correctly. In this case, you win 100% of the time. On the other hand if you guess incorrectly, then you're no worse off than before; your chance of winning is still 50%o. Combining these two cases, your overall chance of winning is better than 50%! Informal arguments about probability, like this one, often sound plausible, but do not hold up under close scrutiny. In contrast, this argument sounds completely implausible- but is actually correct 2.3.2 Anal f the winning Strategy For generality, suppose that I can choose numbers from the set O,1,...,n). Call the lowe number L and the higher number HRandom Variables 9 2.3 The Numbers Game Let’s play a game! I have two envelopes. Each contains an integerin the range 0, 1, . . . , 100, and the numbers are distinct. To win the game, you must determine which envelope con￾tains the larger number. To give you a fighting chance, I’ll let you peek at the number in one envelope selected at random. Can you devise a strategy that gives you a better than 50% chance of winning? For example, you could just pick an evelope at random and guess that it contains the larger number. But this strategy wins only 50% of the time. Your challenge is to do better. So you might try to be more clever. Suppose you peek in the left envelope and see the number 12. Since 12 is a small number, you might guess that that other number is larger. But perhaps I’m sort of tricky and put small numbers in both envelopes. Then your guess might not be so good! An important point here is that the numbers in the envelopes may not be random. I’m picking the numbers and I’m choosing them in a way that I think will defeat your guessing strategy. I’ll only use randomization to choose the numbers if that serves my end: making you lose! 2.3.1 Intuition Behind the Winning Strategy Amazingly, there is a strategy that wins more than 50% of the time, regardless of what numbers I put in the envelopes! Suppose that you somehow knew a number x between my lower number and higher numbers. Now you peek in an envelope and see one or the other. If it is bigger than x, then you know you’re peeking at the higher number. If it is smaller than x, then you’re peeking at the lower number. In other words, if you know an number x between my lower and higher numbers, then you are certain to win the game. The only flaw with this brilliant strategy is that you do not know x. Oh well. But what if you try to guess x? There is some probability that you guess correctly. In this case, you win 100% of the time. On the other hand, if you guess incorrectly, then you’re no worse off than before; your chance of winning is still 50%. Combining these two cases, your overall chance of winning is better than 50%! Informal arguments about probability, like this one, often sound plausible, but do not hold up under close scrutiny. In contrast, this argument sounds completely implausible— but is actually correct! 2.3.2 Analysis of the Winning Strategy For generality, suppose that I can choose numbers from the set {0, 1, . . . , n}. Call the lower number L and the higher number H