auto-update(nvim): 2025-01-18 02:43:37

documents/by-course/pstat-120a/hw1/main.typ

@@ -7,7 +7,9 @@
   date: "Winter 2025",
 )
 
-1. #[
+#set enum(spacing: 2em)
+
++ #[
     #set enum(numbering: "a)", spacing: 2em)
 
     + #[
@@ -31,7 +33,43 @@
     + #[
         $
           E union F &= (A sect B) union (A sect B') \
-          &=
+          &= (A union A) sect (B union B') \
+          &= A sect Omega \
+          &= A
         $
       ]
   ]
+
++ #[
+    #set enum(numbering: "a)", spacing: 2em)
+
+    + ${15, 25, 35, 45, 51, 53, 55, 57, 59, 65, 75, 85, 95 }$
+    + ${50, 52, 56, 58}$
+    + $emptyset$
+  ]
+
++ #[
+    #set enum(numbering: "a)", spacing: 2em)
+
+    + The sample space is every value of the die (1-6) paired with heads and paired with tails. That is, ${1,2,3,4,5,6} times {H,T}$, with cardinality 12.
+    + There are $12^10$ outcomes.
+    + If no participants roll a 5, then we omit any outcome in our sample space where the die outcome is 5, leaving us with 10 outcomes of the die and coin. Now we have $10^10$ outcomes. If at least 1 person rolls a 5, then we note that this is simply the complement of the previous result. So we have $12^10 - 10^10$ outcomes total.
+  ]
+
++ #[
+    #set enum(numbering: "a)", spacing: 2em)
+
+    + #[
+        The sample space can be represented as a 6-tuple where the position 1-6
+        represents balls numbered 1-6, and the value represents the square it
+        was sent to. So it's
+        $ {{x_1,x_2,x_3,x_4,x_5,x_6} : x_i in {1,2,3,4}}, i = 1,...6 $
+      ]
+    + #[
+        When the balls are indistinguishable, we can instead represent it as
+        4-tuples where the position represents the 1st, 2nd, 3rd, or 4th square,
+        and the value represents how many balls landed. Additionally the sum of all
+        the elements must be 6.
+        $ {{x_1, x_2, x_3, x_4} : x_i >= 0, i = 1,...,6 sum_(j=1)^4 x_j = 6} $
+      ]
+  ]