[數學] Probability

1.) Two basketball teams play a series of game in NBA: the first team to win 4 games is declared the overall winner. Suppose that one team is stronger than the other and wins each game with probability 0.6, independent of the outcomes of the other games. Find the probability that the stronger team wins the series in exactly i games, for i=4,5,6,7. Compare the probability that the stronger team wins with the probability that it would win a 2-out-of 3 series.

2.) A long-distance taxi service owns 4 vehicles. These are of different ages and have different repair records. The probabilities that on any given day, each vehicle will be available for use are 0.95,0.90,0.90 and 0.80. Whether one vehicle is available is independent of whether any vehicle is available. Find the standard deviation of the number of vehicles available for use on a given day.

3.) The lifetime (in hours) X of a certain electrical component follows an exponential distribution with parameter λ=0.001. Three of these components operate independently in a system. The system fails if at least two of the components fail. Find the probability that the system operates for at least 200 hours without failure.

4.) The heights of male students entering the university, X, are normally distributed with a mean of 170.4cm. It is known that about 2.5% of the students are taller than 190cm. What is he probability that the mean height of n=4 randomly selected male students exceeds 175cm?

5.) One thousand independent rolls of a fair die will be made. let X be the number of items that the numbers 1 or 2 appear in the one thousand rolls. compute an approximation to P(300≤X≤400).