Multiple of 2 from 1 to 1000 are 2, 4, 6, 8,…., 1000
Let n be the number of terms of above series.
∴
nth term
=
1000
∴
⇒
2
+
(
n
−
1
)
2
=
1000
⇒
⇒
2
+
2
n
−
2
=
1000
⇒
⇒
2
n
=
1000
⇒
∴
n
=
500
∴
Since, the number of multiple of 2 are 500.
So, the multiple of 9 are 9, 18, 27,....,999
Let m be the number of term in above series.
∴
mth term
=
999
∴
⇒
9
+
(
m
−
1
)
9
=
999
⇒
⇒
9
+
9
m
−
9
=
999
⇒
⇒
9
m
=
999
⇒
∴
m
=
111
∴
Since, the number of multiple of 9 are 111. So, the multiple of 2 and 9 both are 18, 36,..., 990
Let p be the number of terms in above series.
∴
pth term
=
990
∴
⇒
18
+
(
p
−
1
)
18
=
990
⇒
⇒
18
+
18
p
−
18
=
990
⇒
⇒
18
p
=
990
⇒
∴
p
=
990
18
=
55
∴
Since, the number of multiple of 2 and 9 are 55.
∴
Number of miltiple of 2 or 9 = 500 + 111-55 = 556
∴
∴
Required probability
=
n
(
E
)
n
(
S
)
=
556
1000
=
0.556
∴