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Magazine Problem 1
There are two roads from \(\mathrm{A}\) to \(\mathrm{B}\) and two roads from \(\mathrm{B}\) to \(\mathrm{C}\). Each of the four roads is blocked by snow with probability \(p\), independently of the others. Find the probability that there is an open road from \(\mathrm{A}\) to \(\mathrm{B}\) given that there is no open route from \(\mathrm{A}\) to \(\mathrm{C}\). If, in addition, there is a direct road from \(\mathrm{A}\) to \(\mathrm{C}\), this road being blocked with probability \(p\). independently of the others, find the required conditional probability.
Problem 2
Calculate the probability that a hand of 13 cards dealt from a normal shuffled pack of 52 contains. exactly two kings and one ace. What is the probability that it contains exactly one ace given that it contains exactly two kings?
Problem 2
A fair coin is tossed repeatedly. Show that, with probability one, a head tums up sooner or later. Show similarly that any given finite sequence of heads and tails occurs eventually with probability one. Explain the connection with Murphy's Law.
Problem 7
Families. Jane has three children, each of which is equally likely to be a boy or a girl independently of the others. Define the events: $$ \begin{aligned} &A=(\text { all the children are of the same sex) } \\ &B=(\text { there is at most one boy } 1 . \\ &C=\\{\text { the family includes a boy and a girl }) \end{aligned} $$ (a) Show that \(A\) is independent of \(B\), and that \(B\) is independent of \(C\). (b) Is \(A\) independent of \(C\) ? (c) Do these results hold if boys and girls are not equally likely? (d) Do these results hold if Jane has four children?
