SAT math question

Test Preparation SAT Preparation
1

<p>Of all the articles in a box, 80% are satisfactory, while 20% are not. The probability of obtaining exactly 5 out of 8 randomly selected articles is:</p>

<p>a) 0.800
b) 0.003
c) 0.147
d) 0.013
e) 0.132</p>

<p>The answer is C</p>

<p>How do you solve this problem?</p>

2

<p>You’ll have to use a binomial distribution to solve this question since the number of satisfactory articles in the box is a discrete random variable.</p>

<p>First, we let K denote the number of satisfactory articles in the box out of the 8 randomly selected articles.
Then, K~B(8,0.8) </p>

<p>Since I’m unable to type out the formula to calculate the probability in a binomial distribution, please refer to this image taken from wikipedia which shows the formula: <a href=“http://upload.wikimedia.org/math/0/3/b/03b6c40b4754fef7a034347da0e479f6.png[/url]”>http://upload.wikimedia.org/math/0/3/b/03b6c40b4754fef7a034347da0e479f6.png&lt;/a&gt;&lt;/p&gt;

<p>In the context of the question, n denotes the number of randomly selected articles, i.e. n=8.
Next, k denotes the value that the variable K will take, so k=5.
Finally, p denotes the probability that 1 randomly selected article is satisfactory, thus p=0.8 (80% chance that the article is satisfactory)</p>

<p>Hence, if you substitute all those values into the formula given, you will obtain the answer Pr(K=5) = 0.147 (corrected to 3 significant figures).</p>

<p>Alternatively, certain graphing calculators may allow you to calculate such probabilities w/o the need for the formula. So if your calculator has that function then that would be great.</p>

3

<p>^^ Thank you so much!</p>

<p>But is that the only method? And can this method be used for ANY type of probability questions?</p>

<p>I am not sure, but i presume that there should be another ‘simpler’ way of solving the question.</p>

4

<p>The hell!
This is not an SAT type of question.</p>

5

<p>ohk :smiley:
that was good relief for me :D</p>

6

<p>Woah, where did you get this problem…</p>

7

<p>^ dont worry, it was a problem from a math sub test 2 book. i just wanted to see whether there was a simpler method of solving the problem than the way it proposed :)</p>

8

<p>Um, using the binomial distribution is the only way I know how to solve this question. I sure hope someone can enlighten me on a simpler method too if it does exist. </p>

<p>But anyway, the binomial distribution cannot be used for any type of probability questions. It can only be used when the following conditions are fulfilled:

  1. The random variable in question (the number of satisfactory articles) is discrete - i.e. it can take a value of a whole number like 5 but not 5.05
  2. The event (in this case, it is the article being satisfactory) is independent of each other, i.e. whether one article is satisfactory has no effect on another being so.
  3. The outcome of each trial is either a success or a failure - i.e. the article can only either be satisfactory or not satisfactory.
  4. the probability of the event is constant, i.e. the probability of the article being satisfactory is constant (80%)</p>

<p>Of course, the above is extra stuff; just for your information. </p>

<p>On another note, I didn’t remember encountering such questions when I took my Maths II subject test in October this year. Only saw one such question in a practice book - forgot which publisher though.</p>

<p>All the best for your test! :)</p>

9

<p>Yeah, binomial distribution is how I solved it. I helped my dad recently with a question like this for a graduate business statistics course. :)</p>

10

<p>Hey, nitcomp, I just thought of a “simpler” way to solve that question. </p>

<p>Basically, the probability of obtaining exactly 5 out of 8 randomly selected articles can be calculated by:</p>

<p>(0.8)(0.8)(0.8)(0.8)(0.8)(0.2)(0.2)(0.2) x [8!/(5! x 3!)]
= (0.8)(0.8)(0.8)(0.8)(0.8)(0.2)(0.2)(0.2) x 56
= 0.147 (3 s.f.)</p>

<p>The first part of the equation is to calculate the probability of obtaining 5 out of 8 randomly selected articles in the order of SSSSSNNN (where S denotes the event that the article is satisfactory while N denotes otherwise). The subsequent part of the equation is the number of different orders in which the articles can be arranged, i.e. there are 56 ways in which you can arrange 5 satisfactory articles and 3 unsatisfactory articles in a row. So, if you multiply both together you’ll obtain the answer. </p>

<p>But I honestly prefer using binomial distribution since I feel that it is quite easy to make careless mistakes or to miscalculate the number of ways which the articles can be arranged using the above method.</p>