Final answer:
To find the probability that both pistons have diameters less than 41mm with a normal distribution (mean = 40mm, standard deviation = 0.5mm), we calculate the z-score for 41mm, find the corresponding probability, and then square it since two pistons are independently selected. The result is approximately 95.5%.
Step-by-step explanation:
The question asks about the probability of selecting two pistons with diameters less than 41mm given that the diameter has a normal distribution with a mean of 40mm and a standard deviation of 0.5mm. To find the probability of one piston being less than 41mm, we need to calculate the z-score and then use the normal distribution table or a calculator to find the corresponding probability. The z-score is given by the formula: Z = (X - µ) / σ.
Where X is the value we are interested in (41mm), µ is the mean (40mm), and σ is the standard deviation (0.5mm). The z-score tells us how many standard deviations the value X is from the mean: Z = (41 - 40) / 0.5 = 2. By looking up the z-score of 2 in a normal distribution table or using a calculator, we find the probability of selecting one piston with a diameter less than 41mm to be approximately 0.9772. To find the probability of both pistons having a diameter of less than 41mm, we multiply the individual probabilities: P(both pistons < 41mm) = P(one piston < 41mm) × P(one piston < 41mm) = 0.9772 × 0.9772 ≈ 0.955. Therefore, the probability that both pistons selected will have diameters less than 41mm is approximately 0.955 or 95.5%.