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**Probability & Statistics in Computer Science**

1. An internet search engine looks for a certain keyword in a sequence of independent websites. It is believed that 20% of the sites contain this keyword. Suppose we select 30 websites randomly. What is the probability that exactly 7 of these websites contain this keyword?

2. Suppose power outages at a university occur at a rate of two per month. What is the probability of at most 5 power outages in any three-month period? Round your answer to three decimal places.

3. Suppose power outages at a university occur at a rate of three per month. What is the probability of at least 7 power outages in any four-month period? Round your answer to three decimal places.

4. Suppose power outages at a university occur at a rate of three per month. What is the probability of fewer than 8 power outages in any four-month period? Round your answer to three decimal places.

5. There are 10 accidents on average at a particular intersection each week. Let the random variable X be the number of accidents at the intersection. Assume independence. Describe the random variable X.

6. According to the U.S. National Center for Health Statistics, there is a 98% chance that a 20-year-old male will survive to the age of 30. Suppose we select 50 males from UTD who are age 20. How many of these students do we expect to live to the age of 30?

Answer :

1. The probability of exactly 7 websites containing the keyword can be calculated using the binomial probability formula.

P(X = 7) = C(30, 7) * (0.2)^7 * (0.8)^(30-7)

where C(30, 7) is the number of combinations of choosing 7 websites out of 30. You can use a calculator or software to calculate the value.

2. To calculate the probability of at most 5 power outages in a three-month period, we can use the Poisson distribution. The rate parameter (λ) is equal to 2 (since there are 2 power outages per month). The probability can be calculated as follows:

P(X ≤ 5) = Σ(k=0 to 5) [e^(-λ) * (λ^k) / k!]

where e is the base of the natural logarithm. You can calculate this probability using a calculator or software.

3. To calculate the probability of at least 7 power outages in a four-month period, we can again use the Poisson distribution. The rate parameter (λ) is equal to 3 (since there are 3 power outages per month). The probability can be calculated as follows:

P(X ≥ 7) = 1 - P(X < 7) = 1 - Σ(k=0 to 6) [e^(-λ) * (λ^k) / k!]

4. To calculate the probability of less than 8 power outages in a four-month period, we can use the Poisson distribution with a rate parameter (λ) of 3. The probability can be calculated as follows:

P(X < 8) = Σ(k=0 to 7) [e^(-λ) * (λ^k) / k!]

5. The random variable X (number of accidents at the intersection) follows a Poisson distribution, given that accidents occur independently and at an average rate of 10 per week.

6. If there is a 98% chance that a 20-year-old male will survive to the age of 30, it means that the probability of survival is 0.98. Out of the 50 males selected from UTD who are at age 20, we can expect approximately 0.98 * 50 = 49 students to live to the age of 30, assuming independence.

learn more about logarithm here:

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