sinclam2/intro-to-probability-solutions

Solutions to the exercises in Grinstead and Snell's Introduction to Probability

Introduction to Probability - Solutions

Solutions to the exercises in Grinstead and Snell's Introduction to Probability

intro-to-probability-solutions's Issues

6.2.7 is incorrect?

I believe 6.2.7 solution is incorrect. It assumes V(X1+X2) = V(X1) + V(X2) which I believe is wrong.

I think the correct solution is:

X - number of heads turn up.
Xi = 1 if head turns up, 0 otherwise on i-th turn.
E[Xi] = 0 * 1/2 + 1 * 1/2 = 1/2
E[X] = E[X1] +E[X2] + E[X3] = 3/2 (also related Theorem 6.3)

E[X^2] = E[(X1+X2+X3)^2] = E[X1^2 + 2 X1 X2 + X2^2 + 2 X1 X3 + 2 X2 X3 + X3^2] =
= E[X1^2] + 2 * E[X1] * E[X2] + E[X2^2] + 2 * E[X1] * E[X3] + 2 * E[X2] * E[X3] + E[X3^2] =
= 1/2 + 2 * 1/2 * 1/2 + 1/2 + 2 * 1/2 * 1/2 + 2 * 1/2 * 1/2 + 1/2 = 3.

V[X] = E[X^2] - E[X]^2 = 3 - (3/2)^2 = 3/4
D[X] = sqrt(3)/2

Your solution to 01.2.09 is incorrect

https://github.com/sinclam2/intro-to-probability-solutions/blob/master/01.2-discrete-probability-distributions/09-Solution.ipynb

I am new to Probability and was working through this solution and was having difficulty coming to the same answer that has been published in various places as a solution to the probability that a student chooses mathematics. Your example is incorrect where you state 5/8 + 5/8 - 1/4 = 10/8 - 5/8. I am not sure how you got the - 5/8 as it should be 1/4, in which case the answer should be 1.

P( Art ∪French)=P(art)+P(French)-P(art ∩French)
P(art ∪French)=5/8+5/8-1/4
=10/8-2/8
=1

There is a high possibility that I am incorrect.....