A batch of 200 phones contains 195 good phones and 5 defective ones. If 2 phones are selected at random, what is the probability of selecting 1 working phone and 1 defective phone?
2 Answers
Explanation:
There are many ways to look at this problem. It's often easiest to consider the possible outcomes and work from there.
If we draw two phones from the box, one at a time, and we don't replace them as we pull them out, there are four possibilities that could happen:
- Pull a good one, then pull a good one again
- Pull a good one, then pull a bad one
- Pull a bad one, then pull a good one
- Pull a bad one, then pull a bad one again
Of those 4 possible situations, two of them satisfy our desired final situation of exactly one being defective: the middle two. We thus have to calculate the two probabilities, and sum them together to get the final result.
A good, then a bad
Assume for this section that
For the first draw, there are 200 phones in the box, of which 195 are good. Thus, the
For the second draw, since we did not replace the good one we pulled the first time, there are now 199 phones in the box, of which 5 are defective. Thus, the
Thus the final probability
A Bad, then a good
This will be similar, but my notation will now have
For the first draw, there are 200 phones in the box, of which 5 are defective. Thus,
For the second draw, there are 199 phones in the box, of which 195 are good. Thus,
Like the last time:
The result is the same! Is this surprising to you?
Final Answer
We now sum both situations to get the final result:
Explanation:
Ways to get "success" (one of each):
The number of ways to select one good phone is
The number of ways to select one bad phone is
Since we can pair each of the 195 good choices with each of the 5 bad choices, the number of ways to select one of each is the product
(We'll ignore simplifying for now.)
Ways to get any pair:
The number of ways to select any unique pair of phones is
Probability of getting one of each:
This is the ratio of
#color(white)= [((195),(1))((5),(1))]/[((200),(2))]" "color(gray)((("notice: " 195 + 5 = 200),(and 1 + 1 = 2))/(("matches with "200),(and 2)))#
#=(195 xx 5)/(((200xx199)/2))#
#=(cancel2 xx cancel195^39 xx cancel5)/(cancel200^4 xx 199)#
#=39/(4 xx 199)" "= 39/796.#