Question

STATEMENT-1 : A card is drawn from a well shuffled ordinary deck of 52 -playing cards. Let \(\mathrm{A}\) be the event that 'card drawn is an Ace' and \(\mathrm{B}\) be the event that 'card drawn is a spade'. Then the events \(A\) and \(B\) are indpendent STATEMENT-2 : Let \(A\) and \(B\) be two non-empty events. If \(P(A / B)=P(A)\), then the events \(A\) and \(B\) are indpendent. (1) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (2) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (3) Statement- 1 is True, Statement-2 is False (4) Statement- 1 is False, Statement- 2 is True

Short answer

Option 1: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Step-by-step solution

Understanding Statement-1

Consider an ordinary deck of 52 playing cards. Let's define two events: \ Event A: Drawing an Ace \ Event B: Drawing a Spade \ The probability of drawing an Ace (\text{A}) is \[ P(A) = \frac{4}{52} = \frac{1}{13} \] \ The probability of drawing a Spade (\text{B}) is \[ P(B) = \frac{13}{52} = \frac{1}{4} \]

Combined Probability of A and B

Next, consider the intersection of events A and B, which is drawing the Ace of Spades. The probability of drawing the Ace of Spades is: \[ P(A \text{ and } B) = \frac{1}{52} \]

Checking for Independence

Two events A and B are independent if and only if \[ P(A \text{ and } B) = P(A) \times P(B) \] \ Substitute in the calculated probabilities: \[ P(A) \times P(B) = \frac{1}{13} \times \frac{1}{4} = \frac{1}{52} \] \ Since \( P(A \text{ and } B) = P(A) \times P(B) \), events A and B are independent. Therefore, Statement-1 is true.

Understanding Statement-2

Statement-2 states that if \(P(A | B) = P(A)\), then A and B are independent. \ Recall that by definition, if events A and B are independent, then the conditional probability \(P(A | B)\) should equal the probability of A, \(P(A)\). Thus, Statement-2 is actually the definition of independent events in probability theory, and therefore Statement-2 is true.

Explanation Verification

Since both Statement-1 and Statement-2 are true, and Statement-2 explains the concept of independence applied in Statement-1, Statement-2 is indeed a correct explanation for Statement-1.

Conclusion

Both Statement-1 and Statement-2 are true, and Statement-2 correctly explains why Statement-1 is true.

Key concepts

probability theory

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes. It helps us measure how probable an event is to occur.

For instance, if we draw a card from a deck of 52 playing cards, we can use probability theory to predict certain results. Let's say we want to know the probability of drawing an Ace. We can calculate this by dividing the number of Aces in the deck (4) by the total number of cards (52), which gives us \( \frac{4}{52} = \frac{1}{13} \).

Probability theory is divided into various parts such as:

• Classical Probability: Based on known outcomes, like a dice roll.
• Empirical Probability: Based on observed data, as in weather forecasting.
• Subjective Probability: Based on personal judgment, like guessing if your favorite team will win.

conditional probability

Conditional probability is the probability of an event occurring given that another event has already happened. It is denoted as \( P(A | B) \) and mathematically defined as: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \textrm{, provided } P(B) > 0 \]

Here's an example. Suppose we want to find out the probability that a drawn card is an Ace, given that it is a Spade. First, we know:

• \( P(A) = \frac{1}{13} \)
• \