Multiple Probability Problems
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1. In a certain location of downtown Baltimore, it costs $7 per day to park at parking lots. An illegally parked car, if caught, will be fined $25, and the chance of being caught is 60%. If money is the only concern of a citizen who must park in this location, should he park at a lot or park illegally?
2. A motorist encounters four consecutive traffic lights, each equally likely to be red or green. Let X be the number of green lights passed by the motorist before being stopped by a red light. What is the PMF of X?
3. A local tavern has 6 bar stools. The bar-tender predicts that if two strangers come into the bar, they will sit in such a way as to leave at least 2 stools between them.
a) if two strangers do come in but choose their seats at random, what is the probability of the bartender's prediction coming true?
b) compute the expected value of the number of stools between the two customers.
4. Suppose a fair die is rolled n times. By using the indicator variable method, find the expected number of the times that a six is followed by at least two other sixes. Now compute the value when n=100.
5. Suppose X can take only the values -1, 0, and 1. If you want to make the mean of X zero and make the variance of X as large as possible, what must be the PMF of X?
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The solution provides answers to multiple probability problems.
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1. In a certain location of downtown Baltimore, it costs $7 per day to park at parking lots. An illegally parked car, if caught, will be fined $25, and the chance of being caught is 60%. If money is the only concern of a citizen who must park in this location, should he park at a lot or park illegally?
Let's calculate the expected money a citizen needs to pay if he parks illegally. If he parks illegally, he might get lucky, with 40% chance, or get fined, with 60% chance. So
E(X)=0.4*0+0.6*25=$15
If he parks in the parking lot, he pays $7.
Since the expected money of illegal parking is higher, he should park at a lot.
2. A motorist encounters four consecutive traffic lights, each equally likely to be red or green. Let X be the number of green lights passed by the motorist before being stopped by a red light. What is the PMF of X?
The probability of encountering red or green light is 0.5. Then X can be 0, 1, 2, 3, and 4.
For X=0, 1, 2, and 3, P(X=i)=, i=0, 1, 2, 3.
In fact, ...
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