Problem 73 - A Complete Resource Book in Mathematics for JEE Main

Question

The probability that certain electronic component fails when first used is \(0.10\). If it does not fail immediately, the probability that it lasts for one year is \(0.99\). The probability that a new component will last for one year is (A) \(0.891\) (B) \(0.692\) (C) \(0.92\) (D) none of these

Step-by-Step Solution

Answer

(A) 0.891

Step 1: Identify the Probabilities Involved

We need to calculate the probability of a new component lasting for one year. Begin by identifying the given probabilities: the probability of the component failing immediately is \(P(F) = 0.10\). If the component does not fail immediately, the probability of it lasting for one year is \(P(L|eg F) = 0.99\), where \(eg F\) denotes the component not failing immediately.

Step 2: Calculate the Complementary Probability

Determine the probability that the component does not fail immediately. This is the complement of failing immediately: \(P(eg F) = 1 - P(F) = 1 - 0.10 = 0.90\).

Step 3: Use the Total Probability Formula

To find the probability that a new component lasts for one year, we need to consider two events: (1) it does not fail immediately and lasts for one year, and (2) it fails immediately (does not contribute to lasting for one year). The probability that it lasts for one year is the product of it not failing immediately and lasting the year: \(P(L) = P(eg F) \cdot P(L|eg F) = 0.90 \times 0.99\).

Step 4: Compute the Final Probability

Calculate the probability that the new component lasts for one year: \(P(L) = 0.90 \times 0.99 = 0.891\).

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring, given that another event has already occurred. To put it simply, it is about narrowing down possible outcomes based on new conditions or information. In our exercise, the probability that the component lasts for one year, given it does not fail immediately, is \(P(L|eg F) = 0.99\).

The symbol \(eg F\) represents the event where the component does not fail immediately. Conditional probability helps us focus on the outcomes specific to our condition, which in this case is that the component has survived the initial use without failing.

• This concept often uses the formula \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), where \(P(A|B)\) is the probability of event A occurring given that B is true.
• This formula is not explicitly used in every scenario but can be insightful when analyzing complex events with dependencies.

Understanding conditional probability allows us to calculate the likelihood of an event by taking into account relevant conditions or assumptions.

Complementary Probability

Complementary probability refers to the probability that an event does not occur. It is essentially the "opposite" chance of something happening. When calculating the complementary probability, we subtract the probability of the event from 1.

For our problem, we need the probability that a component does not fail immediately. Given that the probability of failure is \(P(F) = 0.10\), the complementary probability is \(P(eg F) = 1 - P(F) = 0.90\).

• Complements are easy to calculate and can be very useful in probability theory.
• They provide insight into what we can expect when an event doesn't occur.

In many problems, complementary probabilities simplify calculations by allowing us to focus solely on one part of an event instead of the whole series of possibilities.

Total Probability Formula

The total probability formula connects various probabilities to find the overall probability of a specific event. It is especially helpful when an event can happen through multiple paths or conditions. For this electronic component problem, we need the formula to compute the probability that a new component will last one year.

The formula combines separate probabilities by adding the chances for each potential pathway.

• In our case, the probability that the component lasts one year involves it either surviving the initial use and then lasting or failing immediately and thus not lasting.
• We specifically calculated this as \(P(L) = P(eg F) \cdot P(L|eg F) = 0.90 \times 0.99 = 0.891\).

Using the total probability formula, you break a problem into manageable parts and calculate the likelihood of an overall event, taking into account various ways the event can occur.