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