A box in a certain supply room contains four 40-W lightbulbs, five 60-W bulbs, and six 75-W
bulbs.
i. Suppose that three bulbs are randomly selected. What is the probability that exactly
two of the selected bulbs are rated 75 W?
ii. If two bulbs are randomly selected from the box of lightbulbs and at least one of them
is found to be rated 75 W, what is the probability that both of them are 75-W bulbs?
iii. Given that at least one of the two selected is not rated 75 W, what is the probability
that both selected bulbs have the same rating?
Answers
Answer:
Step-by-step explanation:
Given information:
Number of 40-W bulbs = 3.
Number of 60-W bulbs = 7.
Number of 75-W bulbs = 5.
Probability necessary to examine at least six bulbs to get 75-W bulb is to be found. That is
P
(
X
≥
6
)
is asked to find out.
Total number of bulbs =
3
+
7
+
5
Total number of bulbs = 15.
The formula to find
P
(
X
≥
6
)
is given below.
P
(
X
≥
6
)
=
1
−
P
(
X
<
6
)
P
(
X
≥
6
)
=
1
−
P
(
X
=
1
)
+
P
(
X
=
2
)
+
P
(
X
=
3
)
+
P
(
X
=
4
)
+
P
(
X
=
5
)
Let us find each term separately.
If 75-W bulb is selected on first selection, then it can be chosen in 5 ways.
P
(
X
=
1
)
=
5
15
P
(
X
=
1
)
=
0.3333
If 75-W bulb is selected on second selection, then it can be chosen in
10
×
5
ways.
P
(
X
=
2
)
=
10
×
5
15
×
14
P
(
X
=
2
)
=
0.2381
If 75-W bulb is selected on third selection, then it can be chosen in
10
×
9
×
5
ways.
P
(
X
=
3
)
=
10
×
9
×
5
15
×
14
×
13
P
(
X
=
3
)
=
0.1648
If 75-W bulb is selected on fourth selection, then it can be chosen in ways.
P
(
X
=
4
)
=
10
×
9
×
8
×
5
15
×
14
×
13
×
12
P
(
X
=
4
)
=
0.1099
If 75-W bulb is selected on fifth selection, then it can be chosen in
10
×
9
×
8
×
7
×
5
ways.
P
(
X
=
5
)
=
10
×
9
×
8
×
7
×
5
15
×
14
×
13
×
12
×
11
P
(
X
=
5
)
=
0.0699
Putting these values in the formula, we get,
P
(
X
≥
6
)
=
1
−
(
0.3333
+
0.2381
+
0.1648
+
0.1099
+
0.0699
)
P
(
X
≥
6
)
=
1
−
0.916
P
(
X
≥
6
)
=
0.084