1 If 75 of all graduate students who are admitted to Texas T

1. If 75% of all graduate students who are admitted to Texas Tech actually enroll in graduate programs at Texas Tech, and 40% of all students who apply for admission are admitted, then what percentage of students who apply are admitted and do not enroll at Texas Tech? 2. A balanced coin is tossed times. What is the probability of getting exactly two heads if we are given the information that there were at least two heads in the eight tosses? 3. Consider five dice being rolled, and X equals the sum of the numbers (1 through 6) on the five dice. If the dice are balanced (i.e., equal probabilities of getting 1 through 6 for each die) and independent of each other, find the mean and the variance of X.

Solution

(1) 0.4*(1-0.75) = 0.1

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(2) Given X~Binomial(n=8, p=0.5)

P(X=x)=8Cx*(0.5^8)

P(X=2)=8C2*(0.5^8) = 0.109375

P(X>=2)=1-P(X=0)-P(X=1) =0.9648438

So the probability is

P(X=2|X>=2) = 0.109375/0.9648438

=0.1133603

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(3) mean =E(X)= 17.5

E(X^2) =320.8333

Variance= E(X^2) -[E(X)]^2 = 320.8333 - 17.5^2 =14.5833

Total	Combinations	Probability	x*f(x)	x^2*f(x)
5	1	0.000129	0.000643	0.003215
6	5	0.000643	0.003858	0.023148
7	15	0.001929	0.013503	0.094522
8	35	0.004501	0.036008	0.288066
9	70	0.009002	0.081019	0.729167
10	126	0.016204	0.162037	1.62037
11	205	0.026363	0.289995	3.189943
12	305	0.039223	0.470679	5.648148
13	420	0.054012	0.70216	9.128086
14	540	0.069444	0.972222	13.61111
15	651	0.083719	1.255787	18.83681
16	735	0.094522	1.512346	24.19753
17	780	0.100309	1.705247	28.9892
18	780	0.100309	1.805556	32.5
19	735	0.094522	1.79591	34.1223
20	651	0.083719	1.674383	33.48765
21	540	0.069444	1.458333	30.625
22	420	0.054012	1.188272	26.14198
23	305	0.039223	0.902135	20.7491
24	205	0.026363	0.632716	15.18519
25	126	0.016204	0.405093	10.12731
26	70	0.009002	0.234053	6.085391
27	35	0.004501	0.121528	3.28125
28	15	0.001929	0.054012	1.512346
29	5	0.000643	0.018647	0.540766
30	1	0.000129	0.003858	0.115741
			17.5	320.8333