The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 8.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter less than 4.5 millimeters?
a.0.38, b.0.28, c.0.43, d.0.33
The diameter can be 2.5mm to 8.5mm. There is 6mm of difference. But our target is the ones less than 4.5mm, which is 2mm of difference. Our probability is 2/6 which is the same as 33 percent or .33.
It's a little complicated, sorry I'm not great at explaining things. Do you get it now?
Why did the ball bearing bring a calculator to the manufacturing process? Because it wanted to figure out its probability of being selected!
To find the probability, we need to calculate the area under the probability distribution curve for diameters less than 4.5 millimeters. Since the distribution is uniform, the probability is equal to the width of the interval from 2.5 to 4.5 divided by the total width of the interval from 2.5 to 8.5.
So the probability is (4.5 - 2.5) / (8.5 - 2.5) = 2 / 6 = 1 / 3 ≈ 0.33.
Therefore, the correct answer is d.0.33.
To find the probability that a randomly selected ball bearing has a diameter less than 4.5 millimeters, we need to find the proportion of the uniform distribution that falls within this range.
The formula for a uniform distribution is:
f(x) = 1 / (b - a)
where f(x) is the probability density function, a is the lower bound, and b is the upper bound.
In this case, a = 2.5 millimeters and b = 8.5 millimeters.
Plugging these values into the formula, we get:
f(x) = 1 / (8.5 - 2.5) = 1 / 6 = 0.1667
Now, to find the probability that a random ball bearing has a diameter less than 4.5 millimeters, we need to integrate the probability density function from a to 4.5:
P(X < 4.5) = ∫[2.5, 4.5] f(x) dx
= ∫[2.5, 4.5] 0.1667 dx
= 0.1667 * [x] from 2.5 to 4.5
= 0.1667 * (4.5 - 2.5)
= 0.1667 * 2
= 0.3334
Therefore, the probability that a randomly selected ball bearing has a diameter less than 4.5 millimeters is approximately 0.3334.
Answer: d. 0.33
To find the probability that a randomly selected ball bearing has a diameter less than 4.5 millimeters using a uniform distribution, we need to calculate the area under the probability density function (PDF) curve between 2.5 and 4.5 millimeters.
In this case, the uniform distribution is described by a rectangle with a base of 6 millimeters (8.5 - 2.5) and a height of 1/6 (since it's a uniform distribution).
The area of this rectangle represents the total probability of the random variable falling within the specified range. To find the probability of the diameter being less than 4.5 millimeters, we need to find the ratio of the area of the rectangle between 2.5 and 4.5 millimeters to the total area of the rectangle.
The area of the rectangle between 2.5 and 4.5 millimeters is calculated as follows:
Area = base * height = (4.5 - 2.5) * (1/6) = 2 * (1/6) = 1/3
Now, we need to calculate the total area of the rectangle:
Total Area = base * height = (8.5 - 2.5) * (1/6) = 6 * (1/6) = 1
Finally, we divide the area of the rectangle between 2.5 and 4.5 millimeters by the total area of the rectangle to get the probability:
Probability = (Area between 2.5 and 4.5 millimeters) / (Total area)
Probability = (1/3) / 1 = 1/3 ≈ 0.333
Therefore, the probability that a randomly selected ball bearing has a diameter less than 4.5 millimeters is approximately 0.333.
The correct answer is d. 0.33.