3 numbers x y and z are to bea randomly chosen between 0 and 1.
A What is the probability that X + y + z will be less than 1?
B What is the probability that both x+ y<1 and 1<x+a
Try your best and...
Thanks alot!
A)The volume of the pyramid which is bounded by the planes x=0, y=0, z=0, x+y+z=1 (where 0<=x,y,z<=1 and x+y+z<=1) is equal 1/6.
The volume of the cube 0<=x,y,z<=1 is 1.
Geometric probability P=(1/6)/1=1/6.
B)P(x+y<1)=1/2,
1<x+a what is a?
Thank you and sorry I typed it wrong "a" is y
A) To find the probability that X + Y + Z is less than 1, we need to consider the possible combinations of values that satisfy this condition.
If X + Y + Z is less than 1, then each of X, Y, and Z individually must be less than 1.
Since X, Y, and Z are randomly chosen between 0 and 1, the probability of each being less than 1 is 1.
Therefore, the probability that X + Y + Z is less than 1 is 1.
B) To find the probability that both X + Y < 1 and 1 < X + A, we can consider the two conditions separately and then find the probability that both conditions are satisfied.
1) X + Y < 1:
Since X and Y are randomly chosen between 0 and 1, the probability of their sum being less than 1 is equal to the area of the triangle formed by the points (0,1), (1,0), and (1,1) divided by the area of the square with side length 1 (which represents the possible values for X and Y).
The area of the triangle is (1/2) * 1 * 1 = 1/2.
The area of the square is 1 * 1 = 1.
Therefore, the probability of X + Y < 1 is 1/2.
2) 1 < X + A:
Since X and A are randomly chosen between 0 and 1, the probability of their sum being greater than 1 is equal to the area of the triangle formed by the points (0,1), (1,1), and (1,0) divided by the area of the square with side length 1 (which represents the possible values for X and A).
The area of the triangle is (1/2) * 1 * 1 = 1/2.
The area of the square is 1 * 1 = 1.
Therefore, the probability of 1 < X + A is 1/2.
To find the probability that both conditions are satisfied, we multiply the probabilities of each condition:
Probability of X + Y < 1 and 1 < X + A = (1/2) * (1/2) = 1/4.
Therefore, the probability that both X + Y < 1 and 1 < X + A is 1/4.
To solve this problem, we can break it down into two parts:
Part A: What is the probability that X + Y + Z will be less than 1?
Part B: What is the probability that both X + Y < 1 and 1 < X + A?
Let's solve each part separately:
Part A:
To find the probability that X + Y + Z is less than 1, we need to determine the volume of the region in a three-dimensional space where X + Y + Z < 1.
Since X, Y, and Z are randomly chosen between 0 and 1, they form a unit cube in the three-dimensional space. The volume of this unit cube is 1.
To find the desired region, we need to calculate the volume of the tetrahedron formed by the plane X + Y + Z = 1 and the three coordinate planes X = 0, Y = 0, and Z = 0. The volume of a tetrahedron is given by (1/6) * base area * height.
Here, the base area is the area of the right triangle formed by the three coordinate planes, which is (1/2) * 1 * 1 = 1/2.
The height of the tetrahedron is the perpendicular distance from the plane X + Y + Z = 1 to the origin (0, 0, 0). This distance is 1.
Therefore, the volume of the tetrahedron is (1/6) * (1/2) * 1 = 1/12.
So, the probability that X + Y + Z is less than 1 is the ratio of the volume of the desired region to the volume of the unit cube, which is given by (1/12) / 1 = 1/12.
Part B:
To find the probability that both X + Y < 1 and 1 < X + A, we need to determine the volume of the region that satisfies both these conditions.
The condition X + Y < 1 represents the region below the plane X + Y = 1.
The condition 1 < X + A represents the region above the plane X + A = 1.
To find the desired region, we need to calculate the volume of the region between these two planes. This region is a parallelepiped with a triangular base and height 1.
The base area of the parallelepiped is the area of the right triangle formed by the two planes X + Y = 1 and X + A = 1. This area is (1/2) * 1 * 1 = 1/2.
The height of the parallelepiped is given by the perpendicular distance between the two planes, which is 1.
Therefore, the volume of the parallelepiped is (1/2) * 1 * 1 = 1/2.
Again, the probability is the ratio of the volume of the desired region to the volume of the unit cube, which is given by (1/2) / 1 = 1/2.
So, the probability that both X + Y < 1 and 1 < X + A is 1/2.
In summary:
A) The probability that X + Y + Z is less than 1 is 1/12.
B) The probability that both X + Y < 1 and 1 < X + A is 1/2.