1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Lunna [17]
2 years ago
14

Three numbers in the interval $\left[0,1\right]$ are chosen independently and at random. What is the probability that the chosen

numbers are the side lengths of a triangle with positive area
Mathematics
1 answer:
k0ka [10]2 years ago
5 0

Let a,b,c be the randomly selected lengths. Without loss of generality, suppose a[tex]P(A + B \ge C) = P(A + B - C \ge 0)

where A,B,C are independent random variables with the same uniform distribution on [0, 1].

By their mutual independence, we have

P(A=a,B=b,C=c) = P(A=a) \times P(B=b) \times P(C=c)

so that the joint density function is

P(A=a,B=b,C=c) = \begin{cases}1 & \text{if }(a,b,c)\in[0,1]^3 \\ 0 & \text{otherwise}\end{cases}

where [0,1]^3=[0,1]\times[0,1]\times[0,1] is the cube with vertices at (0, 0, 0) and (1, 1, 1).

Consider the plane

a + b - c = 0

with (a,b,c)\in\Bbb R^3. This plane passes through (0, 0, 0), (1, 0, 1), and (0, 1, 1), and thus splits up the cube into one tetrahedral region above the plane and the rest of the cube under it. (see attached plot)

The point (0, 0, 1) (the vertex of the cube above the plane) does not belong the region a+b-c\ge0, since 0+0-1=-1. So the probability we want is the volume of the bottom "half" of the cube. We could integrate the joint density over this set, but integrating over the complement is simpler since it's a tetrahedron.

Then we have

\displaystyle P(A+B-C\ge0) = 1 - P(A+B-C < 0) \\\\ ~~~~~~~~ = 1 - \int_0^1\int_0^{1-a}\int_{a+b}^1 P(A=a,B=b,C=c) \, dc\,db\,da \\\\ ~~~~~~~~ = 1 - \int_0^1 \int_0^{1-a} (1 - a - b) \, db \, da \\\\ ~~~~~~~~ = 1 - \int_0^1 \frac{(1-a)^2}2\,da \\\\ ~~~~~~~~ = 1 - \frac16 = \boxed{\frac56}

You might be interested in
Estimate a 15% tip on a dinner bill of $81.66 by first rounding the bill amount to the nearest ten dollars.​
julia-pushkina [17]

Answer:

12 dollars

Step-by-step explanation:

Round the bill to the nearest ten dollars

81.66 rounds to 80 dollars

80 * 15%

80 *.15

12

The tip is 12 dollars

3 0
3 years ago
Read 2 more answers
In 1 week of training for the San Francisco Marathon, Jessy Bergeman had training runs of 3.3 miles, 7.7 miles, 5.9 miles, 16.1
GrogVix [38]

Answer:

38.4 miles

Step-by-step explanation:

3.3 + 7.7 + 5.9 + 16.1 + 5.4 = 38.4

7 0
2 years ago
If the integers 23,-9,0-54,31 and 17 are arrange
IgorLugansk [536]

in ascending or descending order?

7 0
3 years ago
Describe how 4 friends could share a sandwich equally
Bezzdna [24]
You can cut it in 4 pieces
3 0
3 years ago
Read 2 more answers
Percy held a gathering where he bought three 0.5-liter water bottles for each guest. If he invited 200 guests, how many gallons
Alexxx [7]
I’m pretty sure it is 100 gallons of water but correct me if i’m wrong.
8 0
3 years ago
Other questions:
  • What answer i need fast
    14·1 answer
  • luisa has 12 playing cards.she place 2 out of every 3 cards face up how many cards are face up and how many are face down
    6·1 answer
  • ((HELP, ASAP! PLEASE EXPLAIN CLEARLY!))
    14·1 answer
  • Expression equivalent to -2.5(4x-3)
    13·1 answer
  • Can someone please help me out with this question? I need an answer ASAP. Thank you!!!
    13·2 answers
  • 10 plus 30 equals what i ont know it
    12·1 answer
  • Mina had 10 pens .She gave 3 pens to her sister . What fraction of pens was<br> given?
    13·1 answer
  • If r/s = 22 and the value of r is 242, what is the value of s
    12·2 answers
  • Help me please , will mark brainliest
    7·2 answers
  • Figure ABCD is reflected over the x-axis to create figure PQRS. What are the coordinates of point Q?​
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!