Add files via upload

ipynb/ProbabilityParadox.ipynb

@@ -2249,7 +2249,7 @@
     }
    },
    "source": [
-    "A table and a plot will give a feel for the `util` function. Notice the characteristics concave-down shape of the plot."
+    "A table and a plot will give a feel for the `util` function. Notice the characterisitc concave-down shape of the plot."
    ]
   },
   {
@@ -3094,7 +3094,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.3"
+   "version": "3.7.0"
   }
  },
  "nbformat": 4,

ipynb/ProbabilitySimulation.ipynb

@@ -485,7 +485,7 @@
     "-  $A$, $B$: the cutoff values they choose: the lower bound of the range of first numbers they will accept.\n",
     "- $a$, $b$: the actual random numbers that appear on the screen.\n",
     "\n",
-    "For example, if player** A** chooses a cutoff of $A$ = 0.6, that means that** A** would accept any first number greater than 0.6, and reject any number below that cutoff. The question is: What cutoff, $A$, should player** A** choose to maximize the chance of winning, that is, maximize P($a$ > $b$)?\n",
+    "For example, if player **A** chooses a cutoff of $A$ = 0.6, that means that **A** would accept any first number greater than 0.6, and reject any number below that cutoff. The question is: What cutoff, $A$, should player **A** choose to maximize the chance of winning, that is, maximize P($a$ > $b$)?\n",
     "\n",
     "First, simulate the number that a player with a given cutoff gets (note that `random.random()` returns a float sampled uniformly from the interval [0..1]):"
    ]
@@ -630,11 +630,11 @@
     "\n",
     "## The Hot New Game Show Problem: Exact Calculation\n",
     "\n",
-    "More promising is the possibility of making `Pwin(A, B)` an exact calculation. But before we get to `Pwin(A, B)`, let's solve a simpler problem: assume that both players** A** and** B** have chosen a cutoff, and have each received a number above the cutoff. What is the probability that** A** gets the higher number? We'll call this `Phigher(A, B)`. We can think of this as a two-dimensional sample space of points in the ($a$, $b$) plane, where$a$ ranges from the cutoff $A$ to 1 and $b$ ranges from the cutoff B to 1. Here is a diagram of that two-dimensional sample space, with the cutoffs $A$=0.5 and $B$=0.6:\n",
+    "More promising is the possibility of making `Pwin(A, B)` an exact calculation. But before we get to `Pwin(A, B)`, let's solve a simpler problem: assume that both players **A** and **B** have chosen a cutoff, and have each received a number above the cutoff. What is the probability that **A** gets the higher number? We'll call this `Phigher(A, B)`. We can think of this as a two-dimensional sample space of points in the ($a$, $b$) plane, where$a$ ranges from the cutoff $A$ to 1 and $b$ ranges from the cutoff B to 1. Here is a diagram of that two-dimensional sample space, with the cutoffs $A$=0.5 and $B$=0.6:\n",
     "\n",
     "<img src=\"http://norvig.com/ipython/probability2da.jpg\" width=413>\n",
     "\n",
-    "The total area of the sample space is 0.5 &times; 0.4 = 0.20, and in general it is (1 - $A$) &middot; (1 - $B$). What about the favorable cases, where**&nbsp;A** beats**&nbsp;B**? That corresponds to the shaded triangle below:\n",
+    "The total area of the sample space is 0.5 &times; 0.4 = 0.20, and in general it is (1 - $A$) &middot; (1 - $B$). What about the favorable cases, where **A** beats **B**? That corresponds to the shaded triangle below:\n",
     "\n",
     "<img src=\"http://norvig.com/ipython/probability2d.jpg\" width=413>\n",
     "\n",
@@ -694,16 +694,16 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We're now ready to tackle the full game. There are four cases to consider, depending on whether**&nbsp;A** and**&nbsp;B** gets a first number that is above or below their cutoff choices:\n",
+    "We're now ready to tackle the full game. There are four cases to consider, depending on whether **A** and **B** gets a first number that is above or below their cutoff choices:\n",
     "\n",
-    "| first $a$ | first $b$ | P($a$, $b$) | P(A wins &vert; $a$, $b$) |  Comment |\n",
+    "| first $a$ | first $b$ | P($a$, $b$) | P(A wins│ $a$, $b$) |  Comment |\n",
     "|:-----:|:-----:| ----------- | ------------- | ------------ |\n",
     "|$a$ > $A$ | $b$ > $B$ | (1 - $A$) &middot; (1 - $B$) | Phigher(*A*, $B$) | Both above cutoff; both keep first numbers |\n",
     "|$a$ < $A$ | $b$ < $B$ | $A$ &middot; $B$ | Phigher(0, 0)  | Both below cutoff, both get new numbers from [0..1] |\n",
-    "|$a$ > $A$ | $b$ < $B$ | (1 - $A$) &middot;  $B$ | Phigher(*A*, 0) |**&nbsp;A** keeps number;**&nbsp;B** gets new number from [0..1] |\n",
-    "|$a$ < $A$ | $b$ > $B$ | $A$ &middot; (1 - $B$) | Phigher(0, $B$) |**&nbsp;A** gets new number from [0..1];**&nbsp;B** keeps number |\n",
+    "|$a$ > $A$ | $b$ < $B$ | (1 - $A$) &middot;  $B$ | Phigher($A$, 0) | **A** keeps number; **B** gets new number from [0..1] |\n",
+    "|$a$ < $A$ | $b$ > $B$ | $A$ &middot; (1 - $B$) | Phigher(0, $B$) | **A** gets new number from [0..1]; **B** keeps number |\n",
     "\n",
-    "For example, the first row of this table says that the event of both first numbers being above their respective cutoffs has probability (1 - $A$) &middot; (1 - $B$), and if this does occur, then the probability of**&nbsp;A** winning is Phigher(*A*, $B$).\n",
+    "For example, the first row of this table says that the event of both first numbers being above their respective cutoffs has probability (1 - $A$) &middot; (1 - $B$), and if this does occur, then the probability of **A** winning is Phigher(*A*, $B$).\n",
     "We're ready to replace the old simulation-based `Pwin` with a new calculation-based version:"
    ]
   },
@@ -936,7 +936,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "What does this [Pringle of Probability](http://fivethirtyeight.com/features/should-you-shoot-free-throws-underhand/) show us? The highest win percentage for**&nbsp;A**, the peak of the surface, occurs when $A$ is around 0.5 and $B$ is 0 or 1. We can confirm that, finding the maximum `Pwin(A, B)` for many different cutoff values of `A` and `B`:"
+    "What does this [Pringle of Probability](http://fivethirtyeight.com/features/should-you-shoot-free-throws-underhand/) show us? The highest win percentage for **A**, the peak of the surface, occurs when $A$ is around 0.5 and $B$ is 0 or 1. We can confirm that, finding the maximum `Pwin(A, B)` for many different cutoff values of `A` and `B`:"
    ]
   },
   {
@@ -1004,7 +1004,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "And what if we run it the other way around, where**&nbsp;B** chooses a cutoff first, and then**&nbsp;A** responds?"
+    "And what if we run it the other way around, where **B** chooses a cutoff first, and then **A** responds?"
    ]
   },
   {
@@ -1033,7 +1033,7 @@
    "metadata": {},
    "source": [
     "In both cases, the rational choice for both players is a cutoff of 0.618034, which corresponds to the  \"saddle point\" in the middle of the plot. This is a *stable equilibrium*; consider fixing $B$ = 0.618034, \n",
-    "and notice that if $A$ changes to any other value, we slip off the saddle to the right or left, resulting in a worse win probability for**&nbsp;A**. Similarly, if we fix $A$ = 0.618034, then if $B$ changes to another value, we ride up the saddle to a higher win percentage for**&nbsp;A**, which is worse for**&nbsp;B**. So neither player will want to move from the saddle point.\n",
+    "and notice that if $A$ changes to any other value, we slip off the saddle to the right or left, resulting in a worse win probability for **A**. Similarly, if we fix $A$ = 0.618034, then if $B$ changes to another value, we ride up the saddle to a higher win percentage for **A**, which is worse for **B**. So neither player will want to move from the saddle point.\n",
     "\n",
     "The moral for continuous spaces is the same as for discrete spaces: be careful about defining your sample space; measure carefully, and let your code take care of the rest."
    ]