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!
Sorry the last one had too many typos...
Three numbers x y and z are to be 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 + z <1 and 1< x+y
Try your best and... Thanks alot!
To find the probabilities, we need to determine the ranges within which the numbers x, y, and z can be chosen. Since the numbers are randomly chosen between 0 and 1, we can assume that they follow a uniform distribution.
A) To find the probability that X + y + z will be less than 1, we need to consider the possible combinations of x, y, and z that satisfy this condition. Let's break it down:
- If x = 0, then the sum of the remaining two numbers (y and z) should be less than 1, which is the entire range (0 to 1).
- If x = 1, then the sum of the remaining two numbers (y and z) should be less than 0, which is not possible.
- For all other cases where 0 < x < 1, the sum of y and z should be less than (1 - x).
Considering these conditions, the probability can be calculated by integrating over the appropriate ranges:
P(X + y + z < 1) = P(x = 0) + ∫[0,1]∫[0, 1-x](1 - x - y)dydx
B) To find the probability that both (x + y < 1) and (1 < x + a), we need to intersect these two conditions. Let's break it down:
- For x + y < 1, we can visualize this as a boundary line in the xy-plane. This line has a slope of -1 and intersects the x-axis at (1, 0).
- For 1 < x + a, we can visualize this as a vertical line in the xa-plane. This line intersects the x-axis at (1, 0).
To calculate the probability, we need to find the area of the region that satisfies both conditions, which is the area below the line x + y = 1 and to the right of the line x = 1.
P((x + y < 1) and (1 < x + a)) = ∫[1,1]∫[0, 1-x](1 - x - y)dydx
Please note that the integration limits may vary depending on the specific range of x, y, and z provided in the problem statement.