JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

Choose the correct answer:

Explanation

Step 1: Define the Total Number of Balls

The urn contains:

Black balls: $10$
White balls: $5$
Total balls: $10 + 5 = 15$

Step 2: Probability of the First Draw

Let $B_1$ be the event that the first ball drawn is black.

The probability of drawing a black ball is:

P(B_1) = \frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{10}{15} = \frac{2}{3}

Step 3: Probability of the Second Draw

Let $B_2$ be the event that the second ball drawn is black. Since the first ball was black and not replaced, the counts change:

Remaining black balls: $10 - 1 = 9$
Remaining total balls: $15 - 1 = 14$

The conditional probability $P(B_2 | B_1)$ is:

P(B_2 | B_1) = \frac{9}{14}

Step 4: Calculate the Combined Probability

The probability that both balls are black is the product of the individual probabilities:

P(B_1 \cap B_2) = P(B_1) \times P(B_2 | B_1)

P(B_1 \cap B_2) = \frac{10}{15} \times \frac{9}{14}

Now, let's simplify the fraction:

P(B_1 \cap B_2) = \frac{2}{3} \times \frac{9}{14}

P(B_1 \cap B_2) = \frac{2 \times 9}{3 \times 14} = \frac{18}{42}

Reducing to the simplest form:

P(B_1 \cap B_2) = \frac{3}{7}

Final Answer

The probability that both drawn balls are black is:

\frac{3}{7}

Choose the correct answer:

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