Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202628 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
28 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Probability

JEE Main 2026 · 28 January · Morning Shift · Mathematics · Probability

A bag contains 10 balls out of which k are red and ( 10-k ) are...

A bag contains 10 balls out of which kk are red and (10−k10-k) are black, where 0≤k≤100 \leq k \leq 10. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

Options

  1. A1455\frac{14}{55}
  2. B755\frac{7}{55}
  3. C7110\frac{7}{110}
  4. D711\frac{7}{11}

Correct answer

A. 1455\frac{14}{55}

Step-by-step solution

Using Bayes' theorem:
P(k=1∣all black)=P(all black∣k=1)⋅P(k=1)P(all black)P(k=1|\text{all black}) = \frac{P(\text{all black}|k=1) \cdot P(k=1)}{P(\text{all black})} where P(k)=111P(k) = \frac{1}{11} for each k∈{0,1,...,10}k \in \{0,1,...,10\}.
When k=1k=1: P(all 3 black∣k=1)=(93)(103)=84120=710P(\text{all 3 black}|k=1) = \frac{\binom{9}{3}}{\binom{10}{3}} = \frac{84}{120} = \frac{7}{10}.
Calculating P(all black)=111∑k=010(10−k3)120P(\text{all black}) = \frac{1}{11}\sum_{k=0}^{10} \frac{\binom{10-k}{3}}{120}.
Numerator is 120+84+56+35+20+10+4+1=330120 + 84 + 56 + 35 + 20 + 10 + 4 + 1 = 330.
So P(all black)=111⋅330120=14P(\text{all black}) = \frac{1}{11} \cdot \frac{330}{120} = \frac{1}{4}.
Therefore P(k=1∣all black)=710⋅11114=710⋅411=28110=1455P(k=1|\text{all black}) = \frac{\frac{7}{10} \cdot \frac{1}{11}}{\frac{1}{4}} = \frac{7}{10} \cdot \frac{4}{11} = \frac{28}{110} = \frac{14}{55}
Practice Probability on Quantrex Academy →

More from Probability

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red… JEE Advanced 2026A candidate has to go to the examination centre to appear in an examination. The… JEE Main 2026A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement… JEE Main 2026A bag contains (N+1) coins - N fair coins, and one coin with 'Head' on both sides. A coin… JEE Main 2026The probabilities that players A and B of a team are selected for the captaincy for a… JEE Main 2026A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the… JEE Main 2026From a month of 31 days, 3 different dates are selected at random. If the probability… JEE Main 2026A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six… JEE Main 2026A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5… JEE Main 2026The probability distribution of a random variable X is given below : (array… JEE Main 2026Let S be a set of 5 elements and P(S) denote the power set of S. Let E be an event of… JEE Main 2026From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one… JEE Main 2026Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls.… JEE Main 2026From the first 100 natural numbers, two numbers first a and then b are selected randomly… JEE Main 2026If a random variable x has the probability distribution array |c|c|c|c|c|c|c|c|c| x & 0 &… JEE Main 2026Two distinct numbers a and b are selected at random from 1,2,3, …, 50 . The probability… JEE Main 2026 Full Probability list Probability JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs