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Alex, Alicia, and Juan fill orders in a fast-food restaurant. Alex incorrectly fills 20% of the orders he takes. Alicia incorrectly fills 12% of the orders she takes. Juan incorrectly fills 5% of the orders he takes. Alex fills 30% of all orders, Alicia fills 45% of all orders, and Juan fills 25% of all orders. An order has just been filled.
1. What is the probability that Alicia filled the order? (Round your answers to 3 decimal places.)
2. Who filled the order is unknown, but the order was filled incorrectly. What are the revised probabilities that Alex, Alicia, or Juan filled the order?
p(Alex) =
p(Alicia) =
p(Juan) =
3. Who filled the order is unknown, but the order was filled correctly. What are the revised probabilities that Alex, Alicia, or Juan filled the order?
p(Alex) =
p(Alicia) =
p(Juan) =
To solve these probability questions, we can use conditional probability. Conditional probability is the probability of an event occurring given that another event has already occurred. Here's how we can approach each question step by step:
1. To find the probability that Alicia filled the order, we need to calculate P(Alicia filling the order | the order was filled). This can be found using Bayes' Theorem:
P(Alicia filling the order | the order was filled) = P(Alicia filling the order) * P(the order was filled | Alicia filled the order) / P(the order was filled)
We are given that P(Alicia filling the order) = 0.45 and P(the order was filled | Alicia filled the order) = 0.12 (since Alicia incorrectly fills 12% of the order she takes).
To find P(the order was filled), we sum the probabilities that each person filled the order and multiply it by the probability that the order was filled correctly. So:
P(the order was filled) = P(Alex filling the order) * P(the order was filled correctly | Alex filled the order) + P(Alicia filling the order) * P(the order was filled correctly | Alicia filled the order) + P(Juan filling the order) * P(the order was filled correctly | Juan filled the order)
We are given that P(Alex filling the order) = 0.30, P(Alicia filling the order) = 0.45, P(Juan filling the order) = 0.25. Also, the order being filled correctly means that each person fills it correctly, so P(the order was filled correctly | Alex filled the order) = (1 - 0.20), P(the order was filled correctly | Alicia filled the order) = (1 - 0.12), P(the order was filled correctly | Juan filled the order) = (1 - 0.05).
Plugging in the given values, we have:
P(the order was filled) = (0.30 * (1 - 0.20)) + (0.45 * (1 - 0.12)) + (0.25 * (1 - 0.05))
Calculate the value of P(the order was filled) and plug it into the first equation to find P(Alicia filling the order).
2. To find the revised probabilities that each person filled the order, given that the order was filled incorrectly, we need to use similar process as in question 1, but we'll use conditional probabilities for incorrect filling instead.
P(Alex filling the order | the order was filled incorrectly) = P(Alex filling the order) * P(the order was filled incorrectly | Alex filled the order) / P(the order was filled incorrectly)
Similarly, we can find P(Alicia filling the order | the order was filled incorrectly) and P(Juan filling the order | the order was filled incorrectly).
To find P(the order was filled incorrectly), we sum the probabilities that each person filled the order incorrectly and multiply it by the probability that the order was filled incorrectly. So:
P(the order was filled incorrectly) = P(Alex filling the order) * P(the order was filled incorrectly | Alex filled the order) + P(Alicia filling the order) * P(the order was filled incorrectly | Alicia filled the order) + P(Juan filling the order) * P(the order was filled incorrectly | Juan filled the order)
Use the given probabilities to calculate P(the order was filled incorrectly) and plug it into the equations for Alex, Alicia, and Juan to find their revised probabilities.
3. To find the revised probabilities that each person filled the order, given that the order was filled correctly, we'll use the same process as in question 2. However, this time, we'll use the conditional probabilities for correct filling instead.
P(Alex filling the order | the order was filled correctly) = P(Alex filling the order) * P(the order was filled correctly | Alex filled the order) / P(the order was filled correctly)
Similarly, we can find P(Alicia filling the order | the order was filled correctly) and P(Juan filling the order | the order was filled correctly).
To find P(the order was filled correctly), we sum the probabilities that each person filled the order correctly and multiply it by the probability that the order was filled correctly. So:
P(the order was filled correctly) = P(Alex filling the order) * P(the order was filled correctly | Alex filled the order) + P(Alicia filling the order) * P(the order was filled correctly | Alicia filled the order) + P(Juan filling the order) * P(the order was filled correctly | Juan filled the order)
Use the given probabilities to calculate P(the order was filled correctly) and plug it into the equations for Alex, Alicia, and Juan to find their revised probabilities.