Create an image representing the theoretical concept behind a complex probability problem concerning mechanical components. Show three mechanical components, distinct from each other, with a transparent background. The first component should be in the shape of a gear with a 95% etched somewhere on it. The second component should be a mechanical lever with 96% imprinted subtly. The third component must be a spindle with 98% marked discreetly. No text should be apparent, merely visual indications of probability. Use colors that are captivating and render the parts clearly distinguishable from one another.

An assembly consists of three mechanical components. Suppose that the probabilities that the first, second, and third components meet specifications are 0.95, 0.96, and 0.98. Assume that the components are independent. Let X be the number of components that meet specifications. Determine the probability mass function of X. Round your answers to five decimal places (e.g. 98.76543).

P(X=0)
P(X=1)
P(X=2)
P(X=3)

P(X = 0) = (0.05)(0.02)(0.01) = 0.00001

P(X = 1) = (0.95)(0.02)(0.01) + (0.05)(0.98)(0.01) +(0.05)(0.02)(0.99) = 0.00167
P(X = 2) = (0.95)(0.98)(0.01) + (0.95)(0.02)(0.99) +(0.05)(0.98)(0.99) =0.07663
P(X = 3) = (0.95)(0.98)(0.99) = 0.92169

Sure, I'll calculate the probabilities for you, but first let me tell you a joke:

Why don't scientists trust atoms?
Because they make up everything!

Now, let's calculate the probabilities:

P(X=0) = (1-0.95) * (1-0.96) * (1-0.98) = 0.00096

P(X=1) = (0.95) * (1-0.96) * (1-0.98) + (1-0.95) * (0.96) * (1-0.98) + (1-0.95) * (1-0.96) * (0.98) = 0.05748

P(X=2) = (0.95) * (0.96) * (1-0.98) + (0.95) * (1-0.96) * (0.98) + (1-0.95) * (0.96) * (0.98) = 0.26928

P(X=3) = (0.95) * (0.96) * (0.98) = 0.8928

So, the probabilities are:
P(X=0) = 0.00096
P(X=1) = 0.05748
P(X=2) = 0.26928
P(X=3) = 0.8928

Remember, probabilities must add up to 1, and rounding them to five decimal places is a serious business.

To determine the probability mass function (PMF) of X, we need to calculate the probabilities for each possible outcome of X.

P(X = 0) represents the probability that none of the components meet specifications. Since the components are independent, we can multiply the individual probabilities of each component not meeting specifications:
P(X = 0) = (1 - 0.95) * (1 - 0.96) * (1 - 0.98) = 0.00624

P(X = 1) represents the probability that exactly one component meets specifications. We can calculate this by multiplying the probability of the one component meeting specifications with the probabilities of the other two components not meeting specifications:
P(X = 1) = (0.95) * (1 - 0.96) * (1 - 0.98) + (1 - 0.95) * (0.96) * (1 - 0.98) + (1 - 0.95) * (1 - 0.96) * (0.98) = 0.07776

P(X = 2) represents the probability that exactly two components meet specifications. We can calculate this by multiplying the probabilities of two components meeting specifications with the probability of one component not meeting specifications:
P(X = 2) = (0.95) * (0.96) * (1 - 0.98) + (0.95) * (1 - 0.96) * (0.98) + (1 - 0.95) * (0.96) * (0.98) = 0.22608

P(X = 3) represents the probability that all three components meet specifications. We can calculate this by multiplying the individual probabilities of each component meeting specifications:
P(X = 3) = (0.95) * (0.96) * (0.98) = 0.87648

Therefore, the probability mass function of X is as follows:
P(X = 0) = 0.00624
P(X = 1) = 0.07776
P(X = 2) = 0.22608
P(X = 3) = 0.87648

To determine the probability mass function (PMF) of X, we need to calculate the probability of each possible outcome for X.

In this case, X represents the number of components that meet specifications. Since each component's probability of meeting specifications is given and the components are assumed to be independent, we can use the binomial distribution to calculate the probabilities.

The formula for the binomial distribution PMF is given by:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Where:
- n is the number of trials (in this case, the number of components)
- k is the number of successful outcomes (in this case, the number of components meeting specifications)
- p is the probability of success (in this case, the probability that a component meets specifications)

Let's calculate each probability:

P(X = 0):
Here, k = 0, n = 3, and p = 0.95 * 0.96 * 0.98 (since all three components need to fail)
P(X = 0) = (3 choose 0) * (0.95 * 0.96 * 0.98)^0 * (1 - (0.95 * 0.96 * 0.98))^(3-0)

P(X = 1):
Here, k = 1, n = 3, and p = 0.95 * 0.96 * 0.98
P(X = 1) = (3 choose 1) * (0.95 * 0.96 * 0.98)^1 * (1 - (0.95 * 0.96 * 0.98))^(3-1)

P(X = 2):
Here, k = 2, n = 3, and p = 0.95 * 0.96 * 0.98
P(X = 2) = (3 choose 2) * (0.95 * 0.96 * 0.98)^2 * (1 - (0.95 * 0.96 * 0.98))^(3-2)

P(X = 3):
Here, k = 3, n = 3, and p = 0.95 * 0.96 * 0.98
P(X = 3) = (3 choose 3) * (0.95 * 0.96 * 0.98)^3 * (1 - (0.95 * 0.96 * 0.98))^(3-3)

Now, let's calculate each probability:

P(X = 0) = (3 choose 0) * (0.95 * 0.96 * 0.98)^0 * (1 - (0.95 * 0.96 * 0.98))^(3-0)
= 1 * 1 * (1 - (0.95 * 0.96 * 0.98))^3
≈ 1 * 1 * (1 - 0.08848)^3
≈ 1 * 1 * 0.85132
≈ 0.85132 (rounded to 5 decimal places)

P(X = 1) = (3 choose 1) * (0.95 * 0.96 * 0.98)^1 * (1 - (0.95 * 0.96 * 0.98))^(3-1)
= 3 * (0.95 * 0.96 * 0.98) * (1 - (0.95 * 0.96 * 0.98))^2
≈ 3 * 0.87584 * (1 - 0.08848)^2
≈ 3 * 0.87584 * 0.77126
≈ 2.36594 (rounded to 5 decimal places)

P(X = 2) = (3 choose 2) * (0.95 * 0.96 * 0.98)^2 * (1 - (0.95 * 0.96 * 0.98))^(3-2)
= 3 * (0.95 * 0.96 * 0.98)^2 * (1 - (0.95 * 0.96 * 0.98))^1
≈ 3 * 0.811073 * (1 - 0.08848)^1
≈ 3 * 0.811073 * 0.91152
≈ 2.19835 (rounded to 5 decimal places)

P(X = 3) = (3 choose 3) * (0.95 * 0.96 * 0.98)^3 * (1 - (0.95 * 0.96 * 0.98))^(3-3)
= 1 * (0.95 * 0.96 * 0.98)^3 * (1 - (0.95 * 0.96 * 0.98))^0
≈ 1 * 0.735091 * (1 - 0.08848)^0
≈ 0.735091 * 1
≈ 0.73509 (rounded to 5 decimal places)

So, the probability mass function (PMF) of X is as follows:

P(X = 0) ≈ 0.85132
P(X = 1) ≈ 2.36594
P(X = 2) ≈ 2.19835
P(X = 3) ≈ 0.73509