We are asked to find the probability that a randomly selected package comes from Machine B and is properly sealed.

Given Information:

• $P(A) = 0.6$: Probability the package comes from Machine A.
• $P(B) = 0.4$: Probability the package comes from Machine B.
• $P(I) = 0.05$: Probability the package is improperly sealed.
• $P(S) = 0.95$: Probability the package is properly sealed.
• $P(A \cap I) = 0.04$: Probability the package comes from Machine A and is improperly sealed.

To solve:

We are asked to find the conditional probability that the package is properly sealed given that it comes from Machine B, i.e., $P(S \mid B)$.

We can apply the following rule for conditional probability: $P(S \mid B) = \frac{P(S \cap B)}{P(B)}$

We already know $P(B) = 0.4$, but we need to find $P(S \cap B)$, the probability that the package is from Machine B and properly sealed. To find this, we can use the fact that the total probability for being improperly sealed is $0.05$, and it includes contributions from both machines.

Since $P(A \cap I) = 0.04$, this means the remaining $0.01$ of improperly sealed packages must come from Machine B, i.e., $P(B \cap I) = 0.01$.

Thus, the probability that the package is properly sealed from Machine B is: $P(S \cap B) = P(B) - P(B \cap I) = 0.4 - 0.01 = 0.39$

Now, applying the conditional probability formula:

$P(S \mid B) = \frac{0.39}{0.4} = 0.975$

Thus, the probability that a package from Machine B is properly sealed is 0.975 or 97.5%.

Let me know if you need further details or have any questions!

Relative questions:

1. What is the probability that a package from Machine A is improperly sealed?
2. What is the probability that a randomly selected package is improperly sealed, regardless of the machine?
3. If a package is improperly sealed, what is the probability that it came from Machine A?
4. What is the probability that a package from Machine B is improperly sealed?
5. What is the overall probability that a package from either machine is properly sealed?

Tip:

For probability problems involving conditional events, always break the problem down by calculating joint probabilities, and then apply conditional probability formulas to get accurate results.