Question

Две фабрики выпускают одинаковые стекла для автомобильных фар. Первая фабрика выпускает 35% этих стекол, вторая 65%. Первая фабрика выпускает 3% бракованных стекол, а вторая 5%. Найдите вероятность того, что случайно купленное в магазине стекло не окажется бракованным.

Answer 1

Событие А — "стекло окажется бракованным"

$H_1$ - стекло изготовленное первой фабрикой;

$H_2$ - стекло изготовленное второй фабрикой.

$P(H_1)=0.35;\\ P(H_2)=0.65\\ P(A|H_1)=0.03\\ P(A|H_2)=0.05$

Вероятность того, что стекло окажется бракованным, по формуле полной вероятности, равна

$P(A)=P(A|H_1)P(H_1)+P(A|H_2)P(H_2)=0.043$

Вероятность того, что случайно купленное в магазине стекло не окажется бракованным, равна

$\overline{P(A)}=1-P(A)=1-0.043=0.957$

Answer 2

 можно вообще без формул :

Пусть 2 фабрики изготовили вместе  х стекол , тогда число

 хороших стекол , изготовленных  на 1 фабрике равно  

 0,35 ·0,97х = 0,3395х , аналогично для второй 0,65 ·0,95х =

0,6175х ,  общее число хороших стекол , изготовленных на

двух фабриках равно   0,3395х + 0,6175х = 0,957x , а

вероятность того, что случайно купленное в магазине стекло

не окажется бракованным  равна  ( 0,957x) : x = 0,957

Answer 3

Problem Analysis

We are given two factories that produce identical glass for car headlights. The first factory produces 35% of the glass, while the second factory produces 65% of the glass. The first factory has a defect rate of 3%, while the second factory has a defect rate of 5%. We need to find the probability that a randomly purchased glass from a store is not defective.

Solution

To find the probability that a randomly purchased glass is not defective, we need to consider the probabilities of purchasing a glass from each factory and the defect rates of each factory.

Let's calculate the probability step by step:

1. The probability of purchasing a glass from the first factory is 35% (0.35). 2. The probability of purchasing a glass from the second factory is 65% (0.65). 3. The defect rate of the first factory is 3% (0.03). 4. The defect rate of the second factory is 5% (0.05).

To calculate the probability that a randomly purchased glass is not defective, we can use the law of total probability. We need to consider two cases: purchasing a glass from the first factory and purchasing a glass from the second factory.

Let's calculate the probability for each case:

Case 1: Purchasing a glass from the first factory: - Probability of purchasing a glass from the first factory: 35% (0.35). - Probability of the glass being defective: 3% (0.03). - Probability of the glass not being defective: 1 - 3% = 97% (0.97).

Case 2: Purchasing a glass from the second factory: - Probability of purchasing a glass from the second factory: 65% (0.65). - Probability of the glass being defective: 5% (0.05). - Probability of the glass not being defective: 1 - 5% = 95% (0.95).

Now, we can calculate the overall probability of a randomly purchased glass not being defective by considering both cases:

Probability of a randomly purchased glass not being defective = (Probability of purchasing from the first factory * Probability of the glass not being defective from the first factory) + (Probability of purchasing from the second factory * Probability of the glass not being defective from the second factory)

Probability = (0.35 * 0.97) + (0.65 * 0.95)

Let's calculate the final probability:

Probability = (0.35 * 0.97) + (0.65 * 0.95)