If a random variable X follows the Binomial distribution B(33, p) such that
3P(X = 0) = P(X = 1), then the value of P(X = 15)/P(X = 18) - P(X = 16)/P(X = 17) is equal to :

Solution

Maths sir · 4 steps

Get q/p from the given condition. After that, each ratio is just a power of q/p.

1. Find q over p. P(X=0) = q³³ and P(X=1) = 33 p q³². So 3q³³ = 33 p q³² gives 3q = 33p, that is q/p = 11.
2. Binomial coefficients cancel. Since C(n, r) = C(n, n-r), we have C(33, 15) = C(33, 18) and C(33, 16) = C(33, 17). Each ratio therefore keeps only powers of p and q.

   P(X=15)/P(X=18) = p¹⁵q¹⁸/p¹⁸q¹⁵ = (q/p)³, P(X=16)/P(X=17) = p¹⁶q¹⁷/p¹⁷q¹⁶ = q/p

3. Put in numbers. The first ratio is 11³ = 1331 and the second is 11. The difference is 1331 - 11 = 1320.
4. So. The value is 1320, which is option (a).

The idea to keep. When r + s = n in B(n, p), the coefficients C(n, r) and C(n, s) are equal, so P(X=r)/P(X=s) is just (q/p)ˢ⁻ʳ.

Answer from the NTA final answer key. Solution by PrepWiser.