JAMIA 2023 — Mathematics PYQ

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Question

JAMIA 2023 — Mathematics PYQ

JAMIA | Mathematics | 2023

A pair of fair dice is thrown independently 3 times. The probability of getting a score of exactly 9 twice is

Choose the correct answer:

Explanation

1. Probability of a Single Event (Score of 9)

When two fair dice are thrown, the total number of outcomes is:

n(S) = 6 \times 6 = 36

The pairs that result in a sum of 9 are:

(3, 6), (4, 5), (5, 4), (6, 3)

Number of favorable outcomes, $n(E) = 4$ .

The probability of getting a score of 9 in one throw ( $p$ ):

p = \frac{n(E)}{n(S)} = \frac{4}{36} = \frac{1}{9}

The probability of not getting a score of 9 ( $q$ ):

q = 1 - p = 1 - \frac{1}{9} = \frac{8}{9}

2. Binomial Distribution Formula

For $n$ independent trials, the probability of getting exactly $r$ successes is:

P(X = r) = \binom{n}{r} p^r q^{n-r}

Here:

Total trials, $n = 3$
Successes required, $r = 2$
$p = \frac{1}{9}$
$q = \frac{8}{9}$

3. Calculation

Substitute the values into the formula:

P(X = 2) = \binom{3}{2} \left(\frac{1}{9}\right)^2 \left(\frac{8}{9}\right)^{3-2}

Since $\binom{3}{2} = 3$ :

P(X = 2) = 3 \times \frac{1}{81} \times \frac{8}{9}

P(X = 2) = \frac{3 \times 8}{81 \times 9}

P(X = 2) = \frac{24}{729}

Simplify the fraction by dividing numerator and denominator by 3:

P(X = 2) = \frac{8}{243}

Final Answer:

The probability of getting a score of exactly 9 twice is:

\frac{8}{243}

Choose the correct answer:

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