Question

Problem 3. You play a game where you first choose a positive integer flip a fair coinn times. You win a prize if you get exactly 2 heads. How should you choose n to maximize your chance of winning? What is the chance if winning with optimal choice of n? There are two equally good choices for the best n. Find both n and then an

Answer 1

Colutions probability of winning when you choose nisaną (2) n(n) [3)?, n(n-1) (1) nt) apply log on both sides. -> fon)= log (n)(n-1)(+)-1). loqnt login-1-466 + (4) e! is is f 1 - H + ditterentiatior a + c + g 2-1) let us find a for which t' is postive & which ľ is negative =) át a > 0.693 i log(+) =0.693147. for 12=3 => f'so for the trans + logh) >0 => -0.69314770 that means f(n) is increasing function boro n <=3 => S + Thr, <0.693147 for xx4 fco for ny=4 that means fin) is decreasing 42=3 Junction for ny=4 when you chose nazis probability of winning 3(3-1) (+) 3+ - 0.375 . probability of winning 464-1) (2) 4 = 0.375 when you chose n=4 is

we should chose either & or ¢ to maximize chances of winning. probability of winning with an optimal choice et n= 0.375