University of Toronto
Department of Mechanical and Industrial Engineering
MIE236: Probability
(Fall 2022)
Solution:
From the look-up Table A.1 with n = 12 and p = 0.7, we have:
(a)
𝑷(? ≤ 𝑿 ≤ ?)
= 𝑷(𝑿 ≤ ?) − 𝑷(𝑿 ≤ ?)
= ?. ???? − ?. ????
= ?. ????
(b)
𝑷(𝑿 ≤ ?) = ?. ????
(c)
𝑷(𝑿 ≥ ?)
= ? − 𝑷(𝑿 ≤ ?)
= ? − ?. ????
= ?. ????
Problem 3:
An engineering student club lists as its members: 2 mechanical engineering students,
3 civil engineering students, 5 computer science students, and 2 chemical engineering students. If
a representative group of 4 is selected at random to attend a national-wide competition, find the
probability that:
a)
all four programs are represented;
d)
all programs except computer science students are represented.
Solution:
a)
Using the extension of the hypergeometric distribution, we have:
(
?
?
)(
?
?
)(
?
?
)(
?
?
)
(
??
?
)
=
?
??
b)
Using the extension of the hypergeometric distribution, we have:
(
?
?
)(
?
?
)(
?
?
)
(
??
?
)
+
(
?
?
)(
?
?
)(
?
?
)
(
??
?
)
+
(
?
?
)(
?
?
)(
?
?
)
(
??
?
)
=
?
???
Problem 4:
A current of
?
amperes flowing through a resistance of
𝑅
ohms varies according to the probability
distribution
?(𝑖) = {
6𝑖(1 − 𝑖), 0 < 𝑖 < 1,
0, ?????ℎ???.
If the resistance varies independently of the current according to the probability distribution
?(?) = {
2?, 0 < ? < 1,
0, ?????ℎ???,
Find the probability distribution for the power
? = ?
2
𝑅
watts.