A factory makes steel rods and steel tubes. The diameter of a steel rod is normally distributed with mean 3.55cm and standard deviation 0.02cm. The internal diameter of a steel tube is normally distributed with mean 3.60cm and standard deviation 0.02cm. A rod and a tube are selected at random. Find the probability that the rod cannot pass through the tube.
the means of the O.D. of the rods and the I.D. of the tubes are 2.5 sd apart
the two normal bell curves intersect midway between the means
... 1.25 s.d. above the rod O.D. and 1.25 s.d. below the tube I.D.
the tubes below this intersection are too small for the corresponding rods
the rods above this intersection are too big for the corresponding tubes
find the areas under the "non-fitting" sections to find the probability
... fraction of rods above 1.25 s.d. AND fraction of tubes below 1.25 s.d.
To find the probability that the rod cannot pass through the tube, we need to compare the diameter of the rod with the internal diameter of the tube.
Let X be the diameter of the rod, and Y be the internal diameter of the tube. We want to find P(X > Y) - the probability that the diameter of the rod is greater than the internal diameter of the tube.
Given that the diameter of the rod is normally distributed with mean 3.55cm and standard deviation 0.02cm, we can use the standard normal distribution to find the z-score for X:
z_x = (X - mean_x) / standard_deviation_x
z_x = (X - 3.55) / 0.02
Similarly, for Y, the internal diameter of the tube, with mean 3.60cm and standard deviation 0.02cm, we can find the z-score for Y:
z_y = (Y - mean_y) / standard_deviation_y
z_y = (Y - 3.60) / 0.02
Since we want to find P(X > Y), we can rewrite it as P(X - Y > 0).
Now, let's standardize the difference between X and Y with the z-scores:
z = (X - Y - (mean_x - mean_y)) / sqrt(standard_deviation_x^2 + standard_deviation_y^2)
z = (X - Y - (3.55 - 3.60)) / sqrt(0.02^2 + 0.02^2)
z = (X - Y + 0.05) / sqrt(0.0008)
Now we have transformed the problem to finding the probability P(z > 0), where z follows a standard normal distribution.
Using a standard normal distribution table or a calculator, we can find the probability P(z > 0).
P(z > 0) is equivalent to finding the area under the curve to the right of 0. Since the standard normal distribution is symmetric, this probability is 0.5.
Therefore, the probability that the rod cannot pass through the tube is 0.5.