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Introduction to Probability by Charles M. Grinstead of Swarthmore College and J. Laurie Snell of Dartmouth College


Exercises 1.1Edit

Problem 1Edit

H or T P(H)-1/2 H-1/2T
1 0.5 0.5
0 0 0
0 -0.16667 -0.5
1 0 0
0 -0.1 -0.5
1 0 0
1 0.071429 0.5
1 0.125 1
0 0.055556 0.5
1 0.1 1
1 0.136364 1.5
0 0.083333 1
1 0.115385 1.5
0 0.071429 1
1 0.1 1.5
1 0.125 2
0 0.088235 1.5
0 0.055556 1
0 0.026316 0.5
1 0.05 1
1 0.071429 1.5
0 0.045455 1
0 0.021739 0.5
1 0.041667 1
0 0.02 0.5
1 0.038462 1
1 0.055556 1.5
1 0.071429 2
1 0.086207 2.5
1 0.1 3
1 0.112903 3.5
1 0.125 4
0 0.106061 3.5
0 0.088235 3
0 0.071429 2.5
1 0.083333 3
0 0.067568 2.5
0 0.052632 2
0 0.038462 1.5
0 0.025 1
0 0.012195 0.5
1 0.02381 1
0 0.011628 0.5
0 0 0
0 -0.01111 -0.5
1 0 0
0 -0.01064 -0.5
1 0 0
1 0.010204 0.5
0 0 0
0 -0.0098 -0.5
0 -0.01923 -1
0 -0.0283 -1.5
0 -0.03704 -2
0 -0.04545 -2.5
1 -0.03571 -2
1 -0.02632 -1.5
0 -0.03448 -2
1 -0.02542 -1.5
0 -0.03333 -2
1 -0.02459 -1.5
1 -0.01613 -1
0 -0.02381 -1.5
0 -0.03125 -2
0 -0.03846 -2.5
1 -0.0303 -2
0 -0.03731 -2.5
0 -0.04412 -3
0 -0.05072 -3.5
0 -0.05714 -4
0 -0.06338 -4.5
1 -0.05556 -4
0 -0.06164 -4.5
1 -0.05405 -4
0 -0.06 -4.5
1 -0.05263 -4
1 -0.04545 -3.5
0 -0.05128 -4
0 -0.05696 -4.5
1 -0.05 -4
0 -0.05556 -4.5
1 -0.04878 -4
0 -0.05422 -4.5
0 -0.05952 -5
1 -0.05294 -4.5
0 -0.05814 -5
1 -0.05172 -4.5
0 -0.05682 -5
0 -0.0618 -5.5
0 -0.06667 -6
1 -0.06044 -5.5
0 -0.06522 -6
1 -0.05914 -5.5
0 -0.06383 -6
0 -0.06842 -6.5
1 -0.0625 -6
0 -0.06701 -6.5
0 -0.07143 -7
1 -0.06566 -6.5
1 -0.06 -6

Problem 2Edit

Problem 3Edit

Problem 4Edit

Games Percent
10 0.4
100 0.298701
1000 0.196172
10000 0.20511
25000 0.213592
100000 0.207788
1000000 0.211411

Problem 5Edit

0.5 = 1-(215/216)^n
0.5 = (215/216)^n
log(0.5) = n (log(215)-log(216))
149.3729

Problem 6Edit

1000 Spins simulated

-6
-32
-56
-94
-56
-40

Problem 10Edit

Total winnings after 500 simulations = -$243

Problem 12Edit

P(Republican) Sample Size Times Won
0.48 1000 7
0.49 1000 24
0.49 3000 10


Chapter 1.2Edit

Problem 1Edit

P\left(a,b,c\right)=m\left(a\right)+m\left(b\right)+m\left(c\right)=1 P\left(a,b,\bar{c}\right)=m\left(a\right)+m\left(b\right)=\frac{5}{6}

P\left(a,\bar{b},c\right)=m\left(a\right)+m\left(c\right)=\frac{2}{3}

P\left(a,\bar{b},\bar{c}\right)=m\left(a\right)=\frac{1}{2}

P\left(\bar{a},b,c\right)=m\left(b\right)+m\left(c\right)=\frac{1}{2}

P\left(\bar{a},b,\bar{c}\right)=m\left(b\right)=\frac{1}{3}

P\left(\bar{a},\bar{b},c\right)=m\left(c\right)=\frac{1}{6}

P\left(\bar{a},\bar{b},\bar{c}\right)=0

Problem 2Edit

(a) \Omega=\left\{ A\bar{B},\bar{A}B\right\}

(b) \Omega=\left\{ HH,HT,TH,TT\right\}

(c) \Omega=\left\{ 1-1,1-2,...,12-6,12-7\right\} - 7*12 outcomes

(d) \Omega=\left\{ 1,2,3,4,5,6,7,8,9,10\right\}

(e) \Omega=\left\{ A,B,C,D,E,F\right\}

Problem 3Edit

(b) Yes

(c) Yes

(d) Yes

Problem 4Edit

(a) First toss is heads

(b) All heads or all tails

(c) Two heads

(d) At least one tails

Problem 5Edit

Problem 6Edit

6P\left(6\right)=5P\left(5\right)=4P\left(4\right)=3P\left(3\right)=2P\left(2\right)=1P\left(1\right)

m\left(1\right)+2m\left(1\right)+3m\left(1\right)+4m\left(1\right)+5m\left(1\right)+6m\left(1\right)=1

m\left(1\right)=\frac{1}{21},m\left(2\right)=\frac{2}{21},m\left(3\right)=\frac{3}{21},m\left(4\right)=\frac{4}{21},m\left(5\right)=\frac{5}{21},m\left(6\right)=\frac{6}{21}

P\left(2,4,6\right)=\frac{2}{21}+\frac{4}{21}+\frac{6}{21}=\frac{12}{21}

Problem 7Edit

Problem 8Edit

m\left(1\right)+m\left(1\right)+2m\left(1\right)=1

P\left(art\right)=P\left(psychology\right)=\frac{1}{4}

P\left(geology\right)=\frac{1}{2}

Problem 9Edit

Problem 10Edit

Problem 11Edit

Problem 12Edit

Problem 13Edit

Problem 14Edit

Problem 15Edit

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