Question

До каси підприємства надійшли банкноти в пачках з двох банків: 50 пачок з першого і 70 - з другого. Ймовірність допущення помилки касирами першого банку становить 0,15%, другого – 0,2%. Яка ймовірність того, що: а) Навмання вибрану пачку сформовано без помилок; б) Пачку без помилок було сформовано касирами другого банку?

Answer 1

Полный расчет приведен в таблице.
Даю пояснения.
1) Определяем "долю" - вероятность пачки каждого банка среди всех.
р11 = 5/12 = 41,67%
р21 = 7/12 =58,33%
2) Определяем вероятность годных для каждого банка
р12 = 1- 0,15 = 0,85
р22 = 1-0,2 = 0,80
3) Вероятность любой годной - сумма произведений двух вероятностей
р1(А)= 0,4167*0,85=0,3542 - для первого банка
р2(А) = 0,5833*0,8 = 0,4667 - для второго банка
Р(А) = 0,35+0,47 = 0,8208 - всего годных
Задача 2
Вероятность годной от первого банка - по формуле Байеса.
Р2(Б) = 0,3542 / 0,8208 = 0,4315 ~ 43% - из первого банка  -ОТВЕТ
Дополнительно: 0,4315/0,8208 ~ 57% - из второго банка

Answer 2

Calculation of Probabilities

To calculate the probabilities, we need to consider the number of error-free bundles and the total number of bundles received from each bank. Let's calculate the probabilities step by step.

1. Probability of an error-free bundle from the first bank: - The probability of an error-free bundle from the first bank is given as 0.15% or 0.0015. - The number of bundles received from the first bank is 50. - Therefore, the number of error-free bundles from the first bank is 50 * (1 - 0.0015) = 49.925.

2. Probability of an error-free bundle from the second bank: - The probability of an error-free bundle from the second bank is given as 0.2% or 0.002. - The number of bundles received from the second bank is 70. - Therefore, the number of error-free bundles from the second bank is 70 * (1 - 0.002) = 69.86.

Calculation of Probabilities

Now, let's calculate the probabilities as requested:

a) Probability of randomly selecting an error-free bundle: - To calculate this probability, we need to consider the total number of error-free bundles and the total number of bundles received from both banks. - The total number of error-free bundles is 49.925 + 69.86 = 119.785. - The total number of bundles received from both banks is 50 + 70 = 120. - Therefore, the probability of randomly selecting an error-free bundle is 119.785 / 120 = 0.9982.

b) Probability of an error-free bundle being formed by the cashiers of the second bank: - To calculate this probability, we need to consider the number of error-free bundles formed by the cashiers of the second bank and the total number of bundles received from both banks. - The number of error-free bundles formed by the cashiers of the second bank is 69.86. - The total number of bundles received from both banks is 120. - Therefore, the probability of an error-free bundle being formed by the cashiers of the second bank is 69.86 / 120 = 0.5822.

Please note that these calculations are based on the given probabilities and assumptions.