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I've a very basic question about probabilities - ie how to calculate the following. In 1 of the answers I get P whereas the lecture notes give 1-p and I do not understand the reasoning. Thanks in advance.

A trawl through the website HorsesForum.com suggests that a nailed-on steel horseshoe will last about 5 to 8 weeks before it needs to be re-set. Assuming the lifetime of a horseshoe is normally distributed with population mean μ = 7 weeks and standard deviation σ = 1.25 weeks.

A second website ‘The Chronicle of the Horse’ advises so-called ‘glued-on’ horseshoes, that is shoes that are glued rather than nailed-on, may be used in cases of laminitis (aka white line disease in USA) where the hoof is unable to accommodate nails. Gleaning the posts on the website seems to indicate that the population mean lifetime of glued-on shoes is μ = 5 weeks with standard deviation σ = 2.0 weeks.

Question 1: Comparing the information from the two websites: What is the probability that a nailed-on horseshoe will last longer than a glued-on horseshoe, i.e the difference between the lifetimes of the two types of horseshoe is greater than zero.

Question 2: Comparing a sample mean to a population value μ (e.g population mean): The website data used in part (i) was in fact based on a sample of n=5 with ̅− = 7. On the basis of this sample data what is the probability that the mean lifetime μ for (the population of) nailed-on horseshoes is at least (μ =) 9 weeks.

Question 3: Comparing the information based on samples from the two websites: The website data used in part (i) was in fact based on a sample of n=5 with ̅− = 7 while in part (ii) the sample size was n=2 and ̅− = 5. On the basis of these sample sizes what is the probability that a nailed-on horseshoe will last longer than a glued-on horseshoe, i.e the difference between the means of the lifetimes of the two types of horseshoe is greater than zero.