answer：Okay, so I have these two problems to solve related to Alex trying to analyze Jamie's performance in tennis. Let me take them one by one.Starting with the first problem: Alex is using a logistic function to model the probability of winning against different opponents. The formula given is ( P(x) = frac{1}{1 + e^{-kx}} ), where ( x ) is the difference in skill levels, and ( k ) is a positive constant. Alex's skill level is 1500, Jamie's is 1550, and ( k = 0.01 ). I need to find the probability that Alex will win against Jamie.Hmm, okay. So first, I should figure out what ( x ) is. Since ( x ) is the difference in skill levels, and Alex is the one trying to win, I think ( x ) would be Alex's skill minus Jamie's skill. So that would be 1500 - 1550. Let me write that down:( x = 1500 - 1550 = -50 )Wait, so ( x ) is negative. That makes sense because Jamie is better, so Alex is at a disadvantage. Now, plugging this into the logistic function:( P(-50) = frac{1}{1 + e^{-k*(-50)}} )Simplify the exponent:( -k*(-50) = 0.01*50 = 0.5 )So now the equation becomes:( P(-50) = frac{1}{1 + e^{0.5}} )I need to calculate ( e^{0.5} ). I remember that ( e ) is approximately 2.71828. So, ( e^{0.5} ) is the square root of ( e ), which is roughly 1.6487.So plugging that in:( P(-50) = frac{1}{1 + 1.6487} = frac{1}{2.6487} )Now, calculating that division. Let me see, 1 divided by 2.6487. I can approximate this.2.6487 goes into 1 about 0.377 times because 2.6487 * 0.377 ≈ 1. So, approximately 0.377.Wait, let me double-check that multiplication:2.6487 * 0.377:First, 2 * 0.377 = 0.7540.6487 * 0.377 ≈ 0.6487 * 0.3 = 0.1946, plus 0.6487 * 0.077 ≈ 0.0500. So total ≈ 0.1946 + 0.05 = 0.2446Adding to the 0.754: 0.754 + 0.2446 ≈ 0.9986, which is close to 1. So, yes, 0.377 is a good approximation.Therefore, the probability that Alex will win against Jamie is approximately 0.377, or 37.7%.Wait, let me think again. Is ( x ) supposed to be Alex's skill minus Jamie's, or the other way around? Because in the logistic function, if ( x ) is positive, the probability increases, and if ( x ) is negative, the probability decreases. So, since Alex is worse, ( x ) is negative, which makes the exponent positive, so the denominator becomes 1 + e^{positive}, which is larger, so the probability is less than 0.5, which makes sense.So, yeah, 37.7% seems correct.Moving on to the second problem: Jamie's match scores follow a normal distribution with a mean of 80 and a standard deviation of 5. Alex wants to find the probability that Jamie scores more than 90 in a match.Alright, so normal distribution. I need to find ( P(X > 90) ) where ( X ) is normally distributed with ( mu = 80 ) and ( sigma = 5 ).To find this probability, I should standardize the score to a z-score and then use the standard normal distribution table or a calculator.The formula for z-score is:( z = frac{X - mu}{sigma} )Plugging in the numbers:( z = frac{90 - 80}{5} = frac{10}{5} = 2 )So, the z-score is 2. Now, I need to find the probability that Z is greater than 2 in the standard normal distribution.I remember that the total area under the standard normal curve is 1, and the curve is symmetric around 0. The area to the left of z=2 is approximately 0.9772, which I recall from the standard normal table. Therefore, the area to the right of z=2 is 1 - 0.9772 = 0.0228.So, the probability that Jamie scores more than 90 is approximately 2.28%.Wait, let me confirm this. If I look up z=2 in the standard normal table, it should give me the cumulative probability up to z=2, which is indeed 0.9772. So, subtracting that from 1 gives the tail probability beyond z=2, which is 0.0228, or 2.28%.Alternatively, I can use the empirical rule, which states that about 95% of the data lies within two standard deviations of the mean. So, 95% of the scores are between 70 and 90. Therefore, the remaining 5% are split equally in the tails beyond 70 and 90. So, each tail has about 2.5%. But wait, that's for two standard deviations, but in this case, 90 is exactly two standard deviations above the mean. So, actually, the area beyond z=2 is 2.28%, which is slightly less than 2.5%. So, my initial calculation was correct.Therefore, the probability that Jamie scores more than 90 is approximately 2.28%.So, summarizing:1. The probability that Alex will win against Jamie is approximately 37.7%.2. The probability that Jamie scores more than 90 is approximately 2.28%.I think that's it. Let me just make sure I didn't make any calculation errors.For the first problem, ( x = -50 ), ( k = 0.01 ), so exponent is 0.5, ( e^{0.5} ≈ 1.6487 ), so ( 1/(1 + 1.6487) ≈ 1/2.6487 ≈ 0.377 ). Yep, that's correct.For the second problem, z=2, area beyond is 0.0228. Correct.So, I think I did everything right.**Final Answer**1. The probability that Alex will win against Jamie is boxed{0.377}.2. The probability that Jamie scores more than 90 is boxed{0.0228}.

answer：First, I need to calculate the total number of calories Alex burns each week by following the workout plan.Alex does 4 cardio sessions per week, and each session burns 300 calories. So, the weekly calories burned from cardio are 4 multiplied by 300, which equals 1,200 calories.Next, Alex does 3 strength training sessions per week, with each session burning 200 calories. The weekly calories burned from strength training are 3 multiplied by 200, totaling 600 calories.Then, Alex does 2 flexibility sessions per week, and each session burns 100 calories. The weekly calories burned from flexibility exercises are 2 multiplied by 100, which is 200 calories.Adding up all the weekly calories burned from each exercise type: 1,200 (cardio) + 600 (strength training) + 200 (flexibility) equals 2,000 calories per week.Alex's goal is to burn 3,000 calories. To find out how many weeks are needed, I divide the total calories goal by the weekly calories burned: 3,000 divided by 2,000 equals 1.5 weeks.Since Alex can't follow the plan for half a week, I'll round up to the next whole number, which is 2 weeks. Therefore, Alex needs to follow the workout plan for 2 weeks to achieve the goal of burning 3,000 calories.

answer：First, I need to determine the total time spent on the legal battles. The developer was involved in 4 legal cases, each taking the same amount of time to resolve, and the total time spent in court was 12 months.Next, I'll calculate the time allocated for finalizing the game development, which is 3 months, and the time planned for marketing, which is 2 months.Finally, I'll add up all these durations to find the total time from the start of the legal battles to the launch of the game.

answer：First, I need to calculate the increase in value from the fresh coat of paint. A 15% increase on the original value of 20,000 is 0.15 multiplied by 20,000, which equals 3,000. Adding this to the original value brings the new value to 23,000.Next, I'll calculate the increase from replacing the old tires. A 10% increase on the new value of 23,000 is 0.10 multiplied by 23,000, which equals 2,300. Adding this to the current value results in a final value of 25,300.Therefore, after both improvements, the car will be worth 25,300.

answer：First, determine the number of tours the heritage officer leads each week by multiplying the number of tours per day by the number of working days: 3 tours/day × 5 days = 15 tours.Next, calculate the total number of visitors by multiplying the number of tours by the maximum number of visitors per tour: 15 tours × 15 visitors/tour = 225 visitors.Therefore, the heritage officer can guide 225 visitors through the Burren in one week.

answer：First, I need to determine the total number of elections Andrzej Rozenek has participated in. According to the information, he has participated in 3 parliamentary elections and 2 local elections.Adding these together, Andrzej has participated in a total of 3 + 2 = 5 elections.Next, I need to find the total number of elections held in Poland since 1989. The data shows there have been 10 parliamentary elections, 8 presidential elections, and 7 local elections.Adding these together, the total number of elections held in Poland since 1989 is 10 + 8 + 7 = 25 elections.Finally, to find the sum of the two numbers, I add the total elections Andrzej participated in to the total elections held in Poland: 5 + 25 = 30.