Question

Теория вероятности 1. В районе 24 человека обучаются на заочном факультете института, из них 6 - на мехфаке, 12 – на агрофаке, и 6 – на экономфаке. Вероятность успешно сдать все экзамены на предстоящей сессии для студентов медфака равно 0.6, агрофака – 0.76 и экономфака – 0.8. Найти вероятность того, что наудачу взятый студент, сдавший успешно все экзамены, окажется студентом экономфака.

Answer 1

Ответ:

Объяснение:

n = 24 - число студентов

n₁ = 6 - число студентов мехфака

n₂ = 12 - число студентов агрофака

n₃ = 6 - число студентов экономфака

Вероятность выбрать студента мехфака:

P(A₁) = n₁ / n = 6 / 24 = 0,25

Вероятность выбрать студента агрофака:

P(A₂) = n₂ / n = 12 / 24 = 0,50

Вероятность выбрать студента экономфака:

P(A₃) = n₃ / n = 6 / 24 = 0,25

Вероятности сдать экзамены:

p₁ = 0,60

p₂ = 0,76

p₃ = 0,80

Вероятность того, что наудачу взятый  студент, успешно сдавший экзамены, окажется студентом экономфака, найдем по формуле полной вероятности:

P(A) = (P(A₃) · p₃) / (P(A₁)·p₁ + P(A₂)·p₂ + P(A₃)·p₃) =

=   (0,25·0,80) / (0,25·0,60 + 0,50·0,76 +  0,25·0,80) ≈ 0,28

Answer 2

Problem Analysis

We are given the following information: - There are 24 students in total at a distance learning faculty of an institute. - Out of these 24 students, 6 are studying at the Mechanical Engineering faculty (mehfak), 12 are studying at the Agricultural faculty (agrofak), and 6 are studying at the Economics faculty (ekonomfak). - The probability of successfully passing all exams in the upcoming session is 0.6 for students at the Medical faculty, 0.76 for students at the Agricultural faculty, and 0.8 for students at the Economics faculty.

We need to find the probability that a randomly chosen student who has successfully passed all exams is from the Economics faculty.

Solution

To solve this problem, we can use the concept of conditional probability. The probability that a randomly chosen student who has successfully passed all exams is from the Economics faculty can be calculated using the following formula:

P(Economics | Successful) = P(Economics and Successful) / P(Successful)

Where: - P(Economics | Successful) is the probability that a student is from the Economics faculty given that they have successfully passed all exams. - P(Economics and Successful) is the probability that a student is from the Economics faculty and has successfully passed all exams. - P(Successful) is the probability that a student has successfully passed all exams.

Let's calculate these probabilities step by step.

1. Calculate the probability of a student being from the Economics faculty and successfully passing all exams: - P(Economics and Successful) = P(Economics) * P(Successful | Economics) - P(Economics) = 6 / 24 = 0.25 (since there are 6 students at the Economics faculty out of 24 in total) - P(Successful | Economics) = 0.8 (given in the problem statement) - P(Economics and Successful) = 0.25 * 0.8 = 0.2

2. Calculate the probability of a student successfully passing all exams: - P(Successful) = P(Medical) * P(Successful | Medical) + P(Agricultural) * P(Successful | Agricultural) + P(Economics) * P(Successful | Economics) - P(Medical) = 0 (since there are no students from the Medical faculty) - P(Successful | Medical) = 0.6 (given in the problem statement) - P(Agricultural) = 12 / 24 = 0.5 (since there are 12 students at the Agricultural faculty out of 24 in total) - P(Successful | Agricultural) = 0.76 (given in the problem statement) - P(Economics) = 6 / 24 = 0.25 (since there are 6 students at the Economics faculty out of 24 in total) - P(Successful | Economics) = 0.8 (given in the problem statement) - P(Successful) = 0 * 0.6 + 0.5 * 0.76 + 0.25 * 0.8 = 0.38

3. Calculate the probability of a student being from the Economics faculty given that they have successfully passed all exams: - P(Economics | Successful) = P(Economics and Successful) / P(Successful) - P(Economics | Successful) = 0.2 / 0.38 ≈ 0.526

Therefore, the probability that a randomly chosen student, who has successfully passed all exams, is from the Economics faculty is approximately 0.526.

Conclusion

The probability that a randomly chosen student, who has successfully passed all exams, is from the Economics faculty is approximately 0.526.