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Precalculus Mathematics Homework Help
What Is the Probability of Alice Having Classes Every Day?
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[QUOTE="Ray Vickson, post: 4954545, member: 330118"] First: sorry, my previous response was for choosing 6 courses, not 7. But my final objection stands: you need to choose courses, not days. Let's look at the case of choosing 6 courses (which I will do now in another way); the case of choosing 7 course is a bit more involved, as it needs consideration of more cases, but I will leave that up to you to pursue. So (for 6 courses) let p(i,j,k,l,m) = probability of choosing i courses on day 1, j courses on day 2, ..., m courses on day 5. You want [tex] \text{answer} = p(2,1,1,1,1) + p(1,2,1,1,1)+p(1,1,2,1,1)+p(1,1,1,2,1) + p(1,1,1,1,2) [/tex] Now each of these terms is a simple (multi-class) hypergeometric probability: if we have ##N## items, ##N_1## of type 1, ##N_2## of type 2, ..., ##N_r## of type r, then the probability ##p(k_1,k_2, \ldots, k_r)## of choosing ##k_1## items of type 1, ##k_2## items of type 2, ..., ##k_r## items of type r in a sample of ##n## items selected without replacement (and with ##k_1 + k_2 + \cdots + k_r = n## is [tex] p(k_1, k_2, \ldots, k_r) = \frac{C(N_1,k_1)\, C(N_2, k_2)\, \cdots \,C(N_r,k_r)}{C(N,n)} [/tex] Here, ##N = N_1 + N_2 + \cdots + N_r##. See, eg., [url]http://en.wikipedia.org/wiki/Hypergeometric_distribution[/url] (esp., last section) or http://www.epixanalytics.com/modelassist/AtRisk/Model_Assist.htm#Distributions/Discrete_distributions/Multivariate_Hypergeometric.htm So, we have ##p(2,1,1,1,1) = C(6,2) C(6,1)^4/C(30,6) = 432/13195 \doteq 0.03274##. You can quite easily see that all the other terms in the answer are equal to the first one, so [tex] \text{answer} = 5\, p(2,1,1,1,1) = C(5,1) \, p(2,1,1,1,1) = 432/2639 \doteq 0.16370 [/tex] Can you see how to extend this to the case of choosing 7 courses? BTW: you should always check your answers. Had you done so you would have seen that your first answer gave a value of ##\text{ans. 1} = 432/377 \doteq 1.146##, while your second one was ##\text{ans. 2} = 1512/377 \doteq 4.011##. I think "haruspex" has already explained to you where your errors lie, so I am restricting myself to giving a positive answer (what to do), rather than a negative one (what you did wrong). [/QUOTE]
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What Is the Probability of Alice Having Classes Every Day?
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