Transcript 投影片 1

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n!

 
 k  (n  k )!k !
Chapter 1.
14. A circular target of unit radius is divided into four annular zones with
outer radii 1/4, 1/2, 3/4, and 1, respectively. Suppose 10 shots are fired
independently at random into the target.
(a) Compute the probability that at most three shots land in the zone bounded
by the circles of radius 1/2 and radius 1.
(b) If 5 shots land inside the disk of radius 1/2, find the probability that at
least one is in the disk of radius 1/4.
a) The probability that a shot lands in the zone bounded
by the circles of radius 1/2 and radius 1 is
 ( 12 )2 3
1


4
1
3/4
1/2
1/4
Let X represent the number of shots landing in the zone
described above. Then, we want to find P(X  3).
10  0 10 10  1 9 10  2 8 10  3 7
P( X  3)     34   14      43   14      43   14      43   14 
0
1
2
3
Chapter 1.
14. A circular target of unit radius is divided into four annular zones with
outer radii 1/4, 1/2, 3/4, and 1, respectively. Suppose 10 shots are fired
independently at random into the target.
(a) Compute the probability that at most three shots land in the zone bounded
by the circles of radius 1/2 and radius 1.
(b) If 5 shots land inside the disk of radius 1/2, find the probability that at
least one is in the disk of radius 1/4.
b) Let X denote the number of shots landing inside the
disk of radius ½, and Y denote the number of shots
landing inside the disk of radius ¼. We want to find
P(Y > 0 | X = 5).
3
P(Y  0 | X  5)  1  P(Y  0 | X  5)  1   
4
P( X  5, Y  0)  3 
 
P(Y  0 | X  5) 
P( X  5)
4
10  5 5
P( X  5)     14   34 
5
5
5
10 
5
5
P( X  5, Y  0)     163   43 
5
1
3/4
1/2
1/4