A box in a certain supply room contains five 40W lightbulbs

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


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



(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?

Solution

Number of 40-W lightbulbs = X₁ = 5

Number of 60-W lightbulbs = X₂ = 3

Number of 75-W lightbulbs = X₃ = 6

So, total number of bulbs = X₁ + X₂ + X₃ = 5+3+6 = 14

Now, three bulbs are randomly drawn from this sample.

(b)

Now for all the bulbs to be of the same rating, they must be aither from 40-W or 60-W or 75-W.

The number of ways of selecting 3 bulbs from 5 40-W bulbs = ⁵C₃ = 10

The number of ways of selecting 3 bulbs from 3 60-W bulbs = ³C₃ = 1

The number of ways of selecting 3 bulbs from 6 75-W bulbs = ⁶C₃ = 20

So, total possible number of combinations = 10 + 1 + 20 = 31

The sample space consists of all possible number of combinations of 3 bulbs that can be selected from any category. So, the size of sample space is = ¹⁴C₃ = 364

Hence, the probability that all the three bulbs selected are of same rating = 31/364 = 0.08516

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(d)

To examine at least 6 bulbs means that the first 5 bulbs must not be of the 75-W rating.

so, possible number of selections for 5 bulbs that do not have 75-W rating is = ⁸C₅ = 56.

This is because, there are 5 & 3 bulbs with rating of 40-W and 60-W respectively. So, there are total 8 bulbs (5+3) that do not have the 75-W rating and we need 5 out of these.

Now, the sample space for this will contain all possible cases of selecting 5 bulbs from the total of 14 bulbs.

So, size of sample space = ¹⁴C₅ = 2002

Hence, the required probability = 56/2002 = 0.027972