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The Law of Total Probability is a fundamental principle in probability theory that helps us calculate the total probability of an event by considering all possible ways in which the event can occur. In this context, the law was used to find the probability that a customer fills their tank, irrespective of the type of gas they choose.
To apply the law, we first identify all the different scenarios in which a customer could fill their tank:

• Using regular gas and filling the tank.
• Using plus gas and filling the tank.
• Using premium gas and filling the tank.

To find the total probability of a customer filling their tank (\( P(B) \)), we sum up the probabilities of these individual events:\[P(B) = P(A_1) \times P(B|A_1) + P(A_2) \times P(B|A_2) + P(A_3) \times P(B|A_3)\]Breaking it down using given probabilities, we get:\[0.40 \times 0.30 + 0.35 \times 0.60 + 0.25 \times 0.50 = 0.455\]Therefore, there's a \(45.5\% \) chance that a customer will fill their tank.

Bayes' Theorem provides us with a way to update the probability of an event based on new evidence. It's a powerful tool for understanding conditional probability. In this exercise, Bayes' Theorem was employed to determine the likelihood of each type of gas being requested, given that the tank is filled.
The theorem is expressed as:

• \( P(A|B) = \frac{P(A \cap B)}{P(B)} \)

Here, \( P(A|B) \) is the conditional probability of event \( A \) given event \( B \), \( P(A \cap B) \) is the joint probability of both events occurring, and \( P(B) \) is the probability of the new evidence.For this problem, we calculated:

• For regular gas (\( A_1 \)): \( P(A_1|B) = \frac{0.12}{0.455} \approx 0.2637 \), meaning if a customer fills the tank, there is about a 26.37% chance they requested regular gas.
• For plus gas (\( A_2 \)): \( P(A_2|B) = \frac{0.21}{0.455} \approx 0.4615 \), indicating a 46.15% chance.
• For premium gas (\( A_3 \)): \( P(A_3|B) = \frac{0.125}{0.455} \approx 0.2747 \), reflecting a 27.47% chance.

This illustrates how Bayes' Theorem helps refine predictions based on observed data.

Joint Probability refers to the likelihood of two events happening at the same time. In probability calculations, it often involves events that can simultaneously occur. In this exercise, we're particularly interested in the scenario where a customer both chooses a type of gas and fills the tank.
Joint probability is typically calculated by multiplying the probability of the first event by the conditional probability of the second event given the first:\[P(A \cap B) = P(A) \times P(B|A)\]For example, considering plus gas and filling the tank (\( A_2 \cap B \)), we calculated the joint probability as follows:

• Start with the probability that a customer opts for plus gas (\( P(A_2) = 0.35 \)).
• Multiply that by the probability that they fill the tank, given they chose plus gas (\( P(B|A_2) = 0.60 \)).

Thus:\[P(A_2 \cap B) = 0.35 \times 0.60 = 0.21\]This means there is a 21% probability that any given customer will both request plus gas and fill their tank. Joint probability is crucial in assessing the likelihood of combined events.