6.2 Add Basketball (math)

Author: another-guy

README.md

@@ -26,7 +26,8 @@
         * 5.0 [Bitwise Operations Cheat Sheet](https://github.com/another-guy/problem-solving-ts/blob/master/src/cracking-the-coding-interview/5-bit-manipulation/5-0-cheat-sheet.md)
         * 5.1 Insert one bit sequence into another
         * 5.3 [Flip bit to win](https://github.com/another-guy/problem-solving-ts/blob/master/src/cracking-the-coding-interview/5-bit-manipulation/5-3-flip-to-win.cr.md). [[Code Review]](https://codereview.stackexchange.com/questions/194733/find-the-longest-length-of-sequence-of-1-bits-achievable-by-flipping-a-single-bi)
-    * 6 TODO
+    * 6 Math and Logic Puzzles
+        * 6.2 Basketball
     * 7 TODO
     * 8 Recursion and Dynamic Programming
         * 8.1 Triple step (non-recursive)

src/cracking-the-coding-interview/6-math-and-logic-puzzles/6-2-basketball.md

@@ -0,0 +1,27 @@
+# Basketball Game
+
+Let $p$ is the probability of a single successful throw.
+
+Therefore, to win in a two-out-of-three-throws game, one of the following outcomes must happen:
+
+* hit+hit+hit $P_{allThree} = p*p*p = p^3$ ;
+* hit+hit+miss OR hit+miss+hit OR miss+hit+hit $P_{twoOnly} = 3 * [p * p * (1-p)] = 3 p^2 (1-p)$ .
+
+The likelyhood of the win in such game is $p^3 + 3p^2(1-p)$
+
+In order for two-out-of-three-throws to be more appealing, the following inequality must be solved:
+
+| . | . |
+|:--:|:--:|
+| $p^3 + 3p^2(1-p) > p$ | divide by $p$ |
+| $p^2 + 3p(1-p) > 1$ | |
+| $p^2 +3p - 3p^2 - 1 > 0$ | |
+| $-2p^2 + 3p - 1 > 0$ | multiply by $-1$ |
+| $2p^2 - 3p + 1 < 0$ | |
+| $(p - 1)(p - \frac{1}{2}) < 0$ | |
+
+From the graph below it's clear that the inequation holds for range of $0.5 < p < 1$:
+
+[![graph][1]][1]
+
+  [1]: https://i.imgur.com/sl1XcnN.png