Random Variables Your goal is to guess a number a between L and H. To avoid confusing equality cases, you select a at random from among the half-integers But what probability distribution should you use? The uniform distribution turns out to be your best bet. An informal justification is that if I figured out that you were unlikely to pick some number-- say 505- then I'd always put 50 and 51 in the evelopes. Then you'd be unlikely to pick an x between L and H and would have less chance of winning After you ve selected the number z, you peek into an envelope and see some number p. If p>I, then you guess that you're looking at the er number If p<a, then you guess that the other number is larger All that remains is to determine the probability that this strategy succeeds. We can do this with the usual four-step method and a tree diagram Step 1: Find the sample space. You either choose a too low(< L), too high(> h),or just right(L<a< H). Then you either peek at the lower number (p= L)or the higher number (p=h). This gives a total of six possible outcomes peeked at choice of x H L/2n 112 x too low wIn (H-L)2n x just right (H-L)/n p=H (H-L)2n x too high 12p=L In (n-H)/n (n-H)/2n H Step 2: Define events of interest. The four outcomes in the event that you win are marked in the tree diagram Step 3: Assign outcome probabilities. First, we assign edge probabilities. Your guess c is too low with probability L/n, too high with probability(n-H)/n, and just right with probability(H-L)/n. Next, you peek at either the lower or higher number with equal probability. Multiplying along root-to-leaf paths gives the outcome probabilities Step 4: Compute event probabilities. The probability of the event that you win is the� � 10 Random Variables Your goal is to guess a number x between L and H. To avoid confusing equality cases, you select x at random from among the half­integers: 1 1 1 1 , 1 , 2 , . . . , n − 2 2 2 2 But what probability distribution should you use? The uniform distribution turns out to be your best bet. An informal justification is that if I figured out that you were unlikely to pick some number— say 50 1 2 — then I’d always put 50 and 51 in the evelopes. Then you’d be unlikely to pick an x between L and H and would have less chance of winning. After you’ve selected the number x, you peek into an envelope and see some number p. If p > x, then you guess that you’re looking at the larger number. If p < x, then you guess that the other number is larger. All that remains is to determine the probability that this strategy succeeds. We can do this with the usual four­step method and a tree diagram. Step 1: Find the sample space. You either choose x too low (< L), too high (> H), or just right (L < x < H). Then you either peek at the lower number (p = L) or the higher number (p = H). This gives a total of six possible outcomes. x just right 1/2 1/2 1/2 1/2 1/2 1/2 L/n (H−L)/n (n−H)/n choice of x # peeked at result probability win win x too high x too low win lose win lose L/2n L/2n (H−L)/2n (H−L)/2n (n−H)/2n (n−H)/2n p=H p=L p=H p=L p=H p=L Step 2: Define events of interest. The four outcomes in the event that you win are marked in the tree diagram. Step 3: Assign outcome probabilities. First, we assign edge probabilities. Your guess x is too low with probability L/n, too high with probability (n − H)/n, and just right with probability (H − L)/n. Next, you peek at either the lower or higher number with equal probability. Multiplying along root­to­leaf paths gives the outcome probabilities. Step 4: Compute event probabilities. The probability of the event that you win is the