Question

A number X is selected at random from the unit interval. Let the events A and B be:
A = “X differs from 1/2 by more than 1/4”
B = “1 - X is less than 1/2.”
Find the events A∩B, A^c∩B, A∪B

Answer 1

Set A =

$$A=\{|x-1/2|>1/4\}=$$

$$\{x-1/2>1/4\ or\ x-1/2<-1/4\}=$$

$$\{x>3/4\ or\ x<1/4\}=$$

$$(0,\frac{1}{4})\cup (\frac{3}{4},1)$$

Set B =

$$B=\{1-x<\frac{1}{2}\}=\{x>\frac{1}{2}\}=(\frac{1}{2},1)$$

Set A∩B =

$$A\cap B=((0,\frac{1}{4})\cup (\frac{3}{4},1))\cap(\frac{1}{2},1)=$$

$$((0,\frac{1}{4})\cap (\frac{1}{2},1))\cup((\frac{3}{4},1)\cap (\frac{1}{2},1))=$$

$$(\frac{3}{4},1)$$

Set A^c =

$$A^c=((0,\frac{1}{4})\cup (\frac{3}{4},1))^c$$

$$[\frac{1}{4},\frac{3}{4}]$$

Set A^c∩B =

$$[\frac{1}{4},\frac{3}{4}]\cap (\frac{1}{2},1)=(\frac{1}{2},\frac{3}{4}]$$

Set A∪B =

$$((0,\frac{1}{4})\cup (\frac{3}{4},1))\cup (\frac{1}{2},1)=$$

$$(0,\frac{1}{4})\cup((\frac{3}{4},1)\cup (\frac{1}{2},1))=$$

$$(0,\frac{1}{4})\cup(\frac{1}{2},1)$$

P(A∩B) = 1-3/4=1/4, P(A^c∩B) = 3/4-1/2 = 1/4, p(A∪B) = (1/4 - 0) + (1 - 1/2) = 3/4.

Answer:A∩B = (3/4, 1) , A^c∩B = (1/2, 3/4], A∪B = (0, 1/4)∪(1/2, 1).