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Problem 78 Easy Difficulty

In Exercise $64,$ consider the following additional information on credit card usage:
70$\%$ of all regular fill-up customers use a credit card.
50$\%$ of all regular non-fill-up customers use a credit card.
60$\%$ of all mid-grade fill-up customers use a credit card.
50$\%$ of all mid-grade non-fill-up customers use a credit card.
50$\%$ of all premium fill-up customers use a credit card.
40$\%$ of all premium non-fill-up customers use a credit card.
Compute the probability of each of the following events for the next customer to arrive (a tree diagram might help).
(a) $\{$ mid-grade and fill-up and credit card $\}$
(b) $\{$ premium and non-fill-up and credit card $\}$
(c) $\{$ premium and credit card $\}$
(d) $\{$ fill-up and credit card $\}$
(e) $\{$ credit card $\}$
(f) If the next customer uses a credit card, what is the probability that s/he purchased premium gasoline?

Answer

Given: P(A1) =0.40 , P(A2) =0.35, P(A3) =0.25,
P(B/A1) =0.30, P(B/A2) =0.60, P(B/A3) =0.50,
P(C/A1&B) =0.70, P(C/A1&B') =0.50, P(C/A2&B) =0.60, P(C/A2&B') =0.50, P(C/A3&B) =0.50, P(C/A3&B') =0.40

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Video Transcript

{'transcript': "okay for this problem. We have one moment here. Yeah, we are considering a gas station where 40% of customers use regular gas. We call that, uh, event a one that has a probability of 0.4 35% use mid grade gas, so probability of a to equal to 0.35 and 25% use premium gas. So you have a three is equal to 0.25 I also know that of those using regular gas, only 30% filled their tanks. So we know that the probability of B given a one is equal to 0.3. You know that the probability that or of the customers using mid grade gas, 60% filled their tanks probability of be given a to 0.6 and the probability of, or rather 50% of those who get premium gas will fill their tanks The probability of be given a three equal to 0.5 Now for part A. We want to find the probability that the next customer will request mid grade gas and fill the tank. So we want to find the probability a to and be so probability of a two and B is equal to the probability of be given a to times the probability of a to so that is going to be 0.6 times 0.35 that is going to equal 0.21 for Part B. We want to find the probability that a customer will fill their tank, so that is the probability of event be. What we can do is we can use the law of total probability. Here. She tells us that the probability of event be where a one in this case, a one to a three, our exclusive and exhaustive. They have to get regular mid grade or premium and their, uh, you can't get regular and grade or any other combination of them so we can use the We'll have total probability, so probability of be given a one times probability of a one, plus the probability of be given a to times the probability of a to plus the probability of be given a three. I'm the probability of a three, which we should recognize that each of these are the same as that's the same as probability of B and a one so on, so that plugging in what we have there we get 0.3 times 0.4 plus 0.6 times 0.35 plus 0.5 I'm 0.25 something. All that up, we get 0.455 Next for part C, the next customer fills the tank. What is the probability that regular gas is requested mid grade gas or premium gas? So that is the probability of a one given B. And similarity is going to be a probability of a two, given the A three given, but can calculate this using I believe it would be. Think it's Bae's here? Um, yes, Bayes Theorem 1.5 in the text. So using base serum that is going to be equal to the probability of a one times the probability of be given a one divided by the probability of B, which in our case is going to be 0.4 times 0.3, divided by 0.455 using the result from Part B. Um, that gives us a 0.27 doing a little bit of rounding. There than probability of a two given B equal to the probability of a to times probability of E given A to divided by the probability of B that is going to give us 0.35 times 0.6 divided by 0.455 is going to equal 0.46 and, lastly, probability of a three given B is equal to. I'm going to skip writing that general form, but we know what we're looking at. Just walk up the 18th for a threes that's going to be 0.25 times 0.5 divided by 0.455 So that is going to be 0.27"}