Question

A hotel in the Algarve had to offer 1 week of vacation to one of its employees as an Easter gift in a random choice. It is known that 80 people work in this hotel, 41 of whom are Portuguese and 39 are foreign nationals. There are 14 Portuguese men and 23 foreign women. Using what you know about conditional probability, check the probability that the gift was offered to a Portuguese citizen, knowing that it was a woman.

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Frederik

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To find the probability that the gift was offered to a Portuguese citizen, given that the person is a woman, we can use the concept of conditional probability.
Let's denote the event that the gift is offered to a Portuguese citizen as A and the event that the person is a woman as B. We are asked to find the probability of A given B, which can be written as P(A|B).
According to the information provided:
Total employees = 80
Portuguese employees = 41
Foreign employees = 39
Portuguese men = 14
Foreign women = 23
We are interested in finding the probability that the person is both Portuguese and a woman, which is the intersection of the Portuguese employees and the female employees.
The probability of the person being Portuguese and a woman is the probability of A intersect B, which can be calculated as:
P(A ∩ B) = P(A) * P(B|A)
where:
P(A) is the probability of being Portuguese, which is 41/80.
P(B|A) is the probability of being a woman given that the person is Portuguese. We can find this by subtracting the number of Portuguese men from the total number of Portuguese employees and then dividing by the total number of Portuguese employees.
Then, the probability that the gift was offered to a Portuguese citizen, given that the person is a woman, can be found using the formula:
P(A|B) = P(A ∩ B) / P(B)
where:
P(B) is the probability of being a woman, which is the total number of female employees divided by the total number of employees.
Let's calculate these probabilities step by step:
P(A) = 41/80
P(B|A) = (41 - 14) / 41 = 27/41
P(B) = (23 + 14) / 80 = 37/80
Then, plug these values into the formula for conditional probability:
P(A|B) = (P(A) * P(B|A)) / P(B) = ((41/80) * (27/41)) / (37/80)
Simplify this expression:
P(A|B) = (27/80) / (37/80) = 27/37 ≈ 0.3375
So, the probability that the gift was offered to a Portuguese citizen, knowing that it was a woman, is approximately 0.3375 or 33.75%.

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