<ol>
<li>A hotel has five single rooms available, for which six men and three women apply. What is the probability that the rooms will be rented to three men and two women?</li>
</ol>
<p>The answer is 5/9. How to get this?</p>
<ol>
<li>Of all the articles in the box, 80% are satisfactory, while 20% are not. The probability of obtaining exactly five good items out of eight randomly selected articles is...</li>
</ol>
<p>The answer is 0.147. Again, how to figure out this one?</p>
<ol>
<li><p>9C5 = 126 ways to choose five people to assign to rooms (assuming indistinguishable rooms, and each person has equal probability of being selected), and there are (6C3)*(3C2) = 60 ways to choose three men and two women. 60/126 = 10/21, not 5/9. If the rooms are distinguishable we would still obtain 10/21 as top and bottom would be multiplied by 5! = 120.</p></li>
<li><p>This doesn’t seem like a well-written question, because that easily depends on how many articles are in the box. For example if there are 8 satisfactory and 2 bad articles then the probability is zero since we need at least three bad items. Also it does not specify whether we draw items with or without replacement.</p></li>
</ol>
<p>Assuming there are infinitely many articles, or we draw with replacement, then the probability distribution is binomial:</p>
<p>P(5/8 good) = (8C5)(.8)^5 (.2)^3 = .147</p>
<p>Where did you get these problems btw? These two problems don’t appear to be very good in terms of quality.</p>
<p>These problems are from Barron’s and they specifically made the second question ambiguous so the articles in the box are unlimited. Your answers are correct though. @MITer94</p>
<p>I’m not sure why they would make #2 ambiguous, maybe just to annoy the heck out of students. Because the answer could very well be zero.</p>