Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(E \cup F\) and \(E \cap F\).

Let \(E\) and \(F\) be two events that are mutually exclusive, and suppose \(P(E)=.2\) and \(P(F)=.5\). Compute: a. \(P(E \cap F)\) b. \(P(E \cup F)\) c. \(P\left(E^{c}\right)\) d. \(P\left(E^{c} \cap F^{c}\right)\)

The number of cars entering a tunnel leading to an airport in a major city over a period of 200 peak hours was observed, and the following data were obtained: $$\begin{array}{rc} \hline \begin{array}{l} \text { Number of } \\ \text { Cars, } x \end{array} & \begin{array}{c} \text { Frequency of } \\ \text { Occurrence } \end{array} \\ \hline 01000 & 15 \\ \hline \end{array}$$ a. Describe an appropriate sample space for this experiment. b. Find the empirical probability distribution for this experiment.

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

Human blood is classified by the presence or absence of three main antigens (A, B, and Rh). When a blood specimen is typed, the presence of the \(\mathrm{A}\) and/or \(\mathrm{B}\) antigen is indicated by listing the letter \(A\) and/or the letter \(B\). If neither the A nor B antigen is present, the letter \(\mathrm{O}\) is used. The presence or absence of the \(\mathrm{Rh}\) antigen is indicated by the symbols \(+\) or \(-\), respectively. Thus, if a blood specimen is classified as \(\mathrm{AB}^{+}\), it contains the \(\mathrm{A}\) and the \(\mathrm{B}\) antigens as well as the \(\mathrm{Rh}\) antigen. Similarly, \(\mathrm{O}^{-}\) blood contains none of the three antigens. Using this information, determine the sample space corresponding to the different blood groups.

What is the probability that a roulette ball will come to rest on an even number other than 0 or 00 ? (Assume that there are 38 equally likely outcomes consisting of the numbers \(1-36,0\), and 00 .)