1. If 75% of all graduate students who are admitted to Texas Tech actually enrol

Question

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.

Explanation / Answer

(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

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