Triangular Distribution Skills practice Find the shaded areas in each triangle (area = probability) !(!)
!"
𝑓 𝑥 =
= !"×!
!"
!
!"
Area = ×6× = 0.45 !
!"
Area = 1 − 0.45 = 0.55 !
𝑓 𝑥 = !"
!
!
!
!"
Area = ×16×
= 0.7619 (4sf) 𝑓 𝑥 =
!(!)
!"×!"
Area = 1 −
!
!
=
!"
!!"
×5×
!"
!!"
= 0.9256 (4sf) !(!"#)
!"#
𝑓 𝑥! =
𝑓 𝑥! =
=
!"#×!!"
!!"!""
𝑓 𝑥! =
𝑓 𝑥! =
!"#×!"#
!(!")
Area = 1 −
−
!
!
×90×
!
=
!"#$##
!"#
×210×
!
!"#
!"#""
!"#
!"#$#!
= 0.6378 (4sf) Area = −
!
!(!"#)
!"#×!"#
!
!"#
!
!
=
!"#
!"#$##
×320×
!"#
!
!"#
×210×
= 0.3876 (4sf)
!
!"#$##
𝑓 𝑥 =
Area = !
!
!(!"##)
!"#$$×!"#$$
!"
×3800×
!"
=
!"#"$%%
𝑓 𝑥! =
= 0.0380 𝑓 𝑥! =
!"#"$%%
Area = !
!
!×!
!(!)
×5×
= 0.2963 Extension: Can you calculate the value of x? (Excellence) Hints: 0.2 is an area. Perhaps consider similar triangles. !(!)
!×!
!"
!"
=
=
−
!
!"
!"
!"
!
!
×3×
!
!"
Formula approach Area = ½ b h = 0.2 1
2 10 − 𝑥
× 10 − 𝑥 ×
= 0.2 2
6×2.5
10 − 𝑥 !
= 0.2 15
10 − 𝑥 ! = 3 10 − 𝑥 = 3 10 − 3 = 𝑥 = 8.27 (2𝑑. 𝑝. ) Similar Triangles Approach. !
!
Large triangle has base of 2.5, height of !, therefore area of !" !
!"
0.2 ÷ !" means the area scale factor is !" Length scale factor is square root of area scale factor Width of triangle is 2.5 ×
!"
!"
Subtract this from 10 to get x = 8.27