Problem 13 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If \(A\) and \(B\) are two independent events such that \(P\left(A \cap B^{\prime}\right)=3 / 25\) and \(P\left(A^{\prime} \cap B\right)=8 / 25\), then \(P(A)\) is (a) \(1 / 5\) (c) \(2 / 5\) (b) \(3 / 8\) (d) \(4 / 5\)

Step-by-Step Solution

Answer

\(P(A) = \frac{3}{5}\).

Step 1: Understand the Given Information

We are provided with two independent events \(A\) and \(B\). We know \(P\left(A \cap B^{\prime}\right) = \frac{3}{25}\) and \(P\left(A^{\prime} \cap B\right) = \frac{8}{25}\). Our goal is to determine the probability \(P(A)\).

Step 2: Recall the Formula for Independent Events

For independent events, \(P(A \cap B) = P(A) \cdot P(B)\) and \(P(A \cap B^{\prime}) = P(A) \cdot P(B^{\prime})\). We will use this to relate the given probabilities to \(P(A)\) and \(P(B)\).

Step 3: Use Complement Rule

Remember the complement rule: \(P(A^{\prime}) = 1 - P(A)\) and \(P(B^{\prime}) = 1 - P(B)\). These rules will be helpful in expressing \(P(B^{\prime})\) and \(P(A^{\prime})\) in terms of \(P(B)\) and \(P(A)\).

Step 4: Express the Known Probabilities in Terms of \(P(A)\) and \(P(B)\)

Using the independence, \(P(A \cap B^{\prime}) = P(A) \cdot (1-P(B)) = \frac{3}{25}\) and \(P(A^{\prime} \cap B) = (1-P(A)) \cdot P(B) = \frac{8}{25}\).

Step 5: Set Up Equations

We now have two equations: \(P(A) \cdot (1-P(B)) = \frac{3}{25}\) and \((1-P(A)) \cdot P(B) = \frac{8}{25}\). We can solve this system to find \(P(A)\) and \(P(B)\).

Step 6: Solve the First Equation for \(P(B)\) in Terms of \(P(A)\)

From the first equation, \(P(A) \cdot (1-P(B)) = \frac{3}{25}\). Simplifying gives \(1-P(B) = \frac{3}{25P(A)}\) and thus, \(P(B) = 1 - \frac{3}{25P(A)}\).

Step 7: Substitute \(P(B)\) in the Second Equation

From the second equation, \((1-P(A)) \cdot P(B) = \frac{8}{25}\). Substituting \(P(B)\) from Step 6, solve the equation for \(P(A)\).

Step 8: Solve for \(P(A)\)

Substitute and solve for \(P(A)\). From our substitution, after simplifying, we find that \(P(A) = \frac{3}{5}\).

Step 9: Verify the Solution

Check that substituting \(P(A) = \frac{3}{5}\) satisfies both initial conditions. \((1-P(A)) \cdot P(B) = \frac{8}{25}\) holds true with this substitution, confirming the solution is correct.

Key Concepts

Independent Events

In probability, when we talk about events being independent, we mean that the occurrence of one event does not influence or affect the occurrence of another event. For instance, if you flip a coin and roll a die, the results of the coin flip don't change the outcome of the die roll and vice versa.

Mathematically, two events, say event \(A\) and event \(B\), are considered independent if the probability of both events happening together, represented as \(P(A \cap B)\), is equal to the product of their individual probabilities: \(P(A) \cdot P(B)\).

Let's break it down into key points:

• If \(A\) and \(B\) are independent, then knowing \(A\) occurs does not change the likelihood of \(B\) occurring.
• Mathematically, for independent events, \(P(A \cap B) = P(A) \cdot P(B)\).
• This property holds true for any combination of independent events, whether you're considering one event or several.

Complement Rule

The complement rule is a fundamental concept in probability that helps determine the likelihood of an event not happening. If an event happens or does not happen, these two scenarios cover all possible outcomes.

The probability of an event not happening, say the event \(A\), is called the complement of \(A\), denoted as \(A'\). The complement rule is mathematically represented as:\[ P(A') = 1 - P(A) \]

Consider this:

• The sum of the probabilities of an event and its complement is always 1, meaning either the event occurs, or it doesn't—no other possibilities.
• If you know one, you can easily find the other using the formula \(P(A') = 1 - P(A)\).
• For instance, if the probability of it raining tomorrow (event \(A\)) is 0.7, the probability of it not raining (\(A'\)) is 1 - 0.7 = 0.3.

Probability Equations

To solve probability problems, such as finding the probability of independent events or using the complement rule, we often use probability equations. These equations allow us to express known probabilities and set up equations to solve for unknown values.

In the case of the given exercise:

- For independent events \(A\) and \(B\), we know:\[ P(A \cap B') = P(A) \cdot (1 - P(B)) \]
- This equation helps us connect \(P(A)\) with known probability values for other combinations involving \(A\) and \(B\).
- We also set up another equation using the complement rule:\[ P(A') = 1 - P(A) \]

With these tools, you can:

• Express probabilities with equations that relate all known and unknown terms.
• Solve for missing probabilities by substituting one equation into another to find specific values like \(P(A)\) or \(P(B)\).
• Verify your solutions by checking if they satisfy all original conditions provided in the problem.