Question

A box in a certain supply room contains four 40-W lightbulbs, three 60-W bulbs, and five 75-W bulbs. Suppose that three bulbs are randomly selected. (Round your answers to four decimal places.)

(a) What is the probability that exactly two of the selected bulbs are rated 75-W?

(b) What is the probability that all three of the selected bulbs have the same rating?

(c) What is the probability that one bulb of each type is selected?

(d) Suppose now that bulbs are to be selected one by one until a 75-W bulb is found. What is the probability that it is necessary to examine at least six bulbs?

Answer 1

Concepts and reason

Combinations gives us in how many ways a particular event can happen or in how many one can select a sample from the given population.

Probability gives us the chance to get a particular event.

In certain cases, dividing the number of ways obtained using combinations with total number of possible ways gives the probability of that event.

Fundamentals

Combinations:

The number of different arrangements of n objects using of then, in which

1.The n objects are distinct,

2.Once an object is used it cannot be repeated, and

3.The order is not important

The formula for the combination is.

(a)

Number of 40-W lightbulbs = 7

Number of 60-W lightbulbs = 6

Number of 75-W lightbulbs = 5

Total number of lightbulbs = 7 + 6 + 5 = 18

Select 3 bulbs from 18 bulbs using with replacement technique.

Number of ways selecting 3 bulbs from 18 lightbulbs is

Number of way selecting two 75-W lightbulbs and one from remaining bulbs is

Compute the probability that exactly two of the selected bulbs are rated 75-W.

(b)

Compute the probability that all three bulbs have same rating.

(c)

Compute the probability that one bulb of each type is selected.

(d)

Suppose bulbs are to be selected one by one until a 75-watt bulb is found

Then, compute the probability that examine at least 6 bulbs.

Ans: Part a

Probability that exactly two of the selected bulbs are rated 75-W is 0.1593.

Part b

Probability that all three bulbs have same rating is 0.0797.

Part c

Probability that one bulb of each type is selected is 0.2574.

Part d

Probability that examine at least 6 bulbs if the bulbs are to be selected one by one until a 75-watt bulb is found is 0.1502.

Answer 2

There are (12 C 3) = 220 ways to choose 3 bulbs.

a) Exactly two 75W can happen (5C2) = 10 ways.
One non-75W can happen (7C1) = 7 ways.
Combination can happen 10 * 7 = 170 ways.
Probability = 170/220 = .77 or 77%

b) All three the same can happen 3 ways; 40-40-40, 60-60-60, 75-75-75.
(4C3) + (3C3) + (5C3) = 4 + 1 + 10 = 15 ways
Probability = 15 /220 = .07678 or 7.678%

c) One of each can happen (4C1) * (4C1) * (7C1) = 112 ways
Probability = 112 / 560 = .0681 or 6.81%

d) The only way that at least 6 bulbs are required is if the first 5 are chosen from the 7 non-75W bulbs.

There are (12 C 5) = 792 ways to choose the first 5 bulbs.
Of those (7C5) = 21 ways to choose from only the non-75W bulbs
Probability = 21 / 792 = .0265 or 2.65%