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Thelazysandwich
03.07.2020 •
Mathematics
Divide a line segment into two parts by uniformly selecting a point inside the segment. Determine the probability that the longer part is at least 3 times the length of the shorter part. Hint: Use interval [0,L] to denote the line segment. Let X be the dividing point. How to represent the event {the longer part is at least 3 times the length of the shorter part} in terms of X.
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Ответ:
The probability that the longer part is at least 3 times the length of the shorter part is 0.5
Step-by-step explanation:
Let us assume,
X be randomly selected the dividing point in the interval [0,L]. So, X has a uniform distribution with pdf
f(X) = 1/L ( where X is greater or equal than 0 and L is greater or equal than L)
The length of the complete line segment is L and length of the segment to the left of X is X and to the right is L-X
There are 2 cases ,
- The left segment is shorter than the right one. Then the length of right segment (which is L-X) should be at least 3 times the length of the left segment (which is X). So,
L- X < or = to 3X
and then , X < or = L/4
- The left segment is the longer one. Then the length of left segment (which is X) should be at least 3 times the length of the right segment (which is L-X). That is,
3 (L- X ) < or = to X
and then , X > or = L *3/4
We can say that the longer part is at least 3 times the length of the shorter part if
X > or = L *3/4 or X < or = L/4
The probability that the longer part is at least 3 times the length of the shorter part is
P(X > or = L *3/4 U X < or = L/4) = 1 - (X > or = L *3/4 U X < or = L/4)
=![\int\limits^\frac{3}{4L} _\frac{L}{4} f{\frac{1}{L} } \, dx](/tpl/images/0700/8591/ef05f.png)
= 1 - 1/L (3/4L - L/4)
= 1 - 0.5
=0.5
The probability that the longer part is at least 3 times the length of the shorter part is 0.5
Ответ:
A. y=2.5x+47. B. 62 degrees
Step-by-step explanation: