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Probability - Geometric probability

For COMPETITION
Number of Total Problems: 2.
FOR PRINT ::: (Book)

Problem Num : 1

From : AMC10

Type:

Section:Probability

Theme:
Adjustment# : 0

Difficulty: 1

'
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$ , $(4,0)$ , $(4,1)$ , and $(0,1)$ . What is the probability that $x<y$ ?
$mathrm{(A) } frac{1}{8}qquad mathrm{(B) } frac{1}{4}qquad mathrm{(C) } frac{3}{8}qquad mathrm{(D) } frac{1...$
'

Category Geometric probability

Analysis

Solution/Answer
The rectangle has a width of $4$ and a height of $1$ .
The area of this rectangle is $4cdot1=4$ .
The line $x=y$ intersects the rectangle at $(0,0)$ and $(1,1)$ .
The area which $x>y$ is the right isosceles triangle with side length $1$ that has vertices at $(0,0)$ , $(1,1)$ , and $(0,1)$ .
The area of this triangle is $frac{1}{2}cdot1^{2}=frac{1}{2}$
Therefore, the probability that $x<y$ is $frac{frac{1}{2}}{4}=frac{1}{8} Rightarrow A$

Answer:

Problem Num : 2

From : AMC10B

Type:

Section:Probability

Theme:
Adjustment# : 0

Difficulty: 1

'
Two real numbers are selected independently at random from the interval $[-20, 10]$ . What is the probability that the product of those numbers is greater than zero?
$extbf{(A)} frac{1}{9} qquad extbf{(B)} frac{1}{3} qquad extbf{(C)} frac{4}{9} qquad extbf{(D)} frac{5}{9} qq...$
'

Category Geometric probability

Analysis

Solution/Answer
Answer:

Array ( [0] => 7778 [1] => 8151 ) 2