91Ó°ÊÓ

I n t e r n e t ~ I n v e s t m e n t s ~ i n ~ t h e ~ \(90 \mathrm{~s}\) The following excerpt is from an article in The New York Times in July, \(1999 .^{27}\) While statistics are not available for Web entrepreneurs who fail, the venture capitalists that finance such Internet start-up companies have a rule of thumb. For every 10 ventures that receive financing - and there are plenty who do not- 2 will be stock market successes, which means spectacular profits for early investors; 3 will be sold to other concerns, which translates into more modest profits; and the rest will fail. a. What is a sample space for the scenario? b. Write down the associated probability distribution. c. What is the probability that a start-up venture that receives financing will realize profits for early investors?

Each day, there is a \(40 \%\) chance that you will sell an automobile. You know that \(30 \%\) of all the automobiles you sell are two-door models, and the rest are four-door models.

Weather Prediction There is a \(50 \%\) chance of rain today and a \(50 \%\) chance of rain tomorrow. Assuming that the event that it rains today is independent of the event that it rains tomorrow, draw a tree diagram showing the probabilities of all outcomes. What is the probability that there will be no rain today or tomorrow?

Weather Prediction There is a \(20 \%\) chance of snow today and a. \(20 \%\) chance of snow tomorrow. Assuming that the event that it snows today is independent of the event that it snows tomorrow, draw a tree diagram showing the probabilities of all outcomes. What is the probability that it will snow by the end of tomorrow?

\(\nabla\) True or false? Every sample space \(S\) is a finite set. Explain.

\(\nabla\) Two distinguishable dice are rolled. Could there be two mutually exclusive events that both contain outcomes in which the numbers facing up add to 7 ?

If \(80 \%\) of candidates for the soccer team pass the fitness test, and only \(20 \%\) of all athletes are soccer team candidates who pass the test, what percentage of the athletes are candidates for the soccer team?

Find a formula for the probability of the union of three (not necessarily mutually exclusive) events \(A, B\), and \(C\).