A bag contains 4 red, 5 blue and 3 green balls. If two balls are drawn at random from the bag, then which of the following statements are correct?
(A) The probability that both balls are red is \(\frac{1}{11}\).
(B) The probability that one ball is red, and one ball is blue is \(\frac{10}{33}\).
(C) The probability that both balls are blue is \(\frac{5}{33}\).
(D) The probability that both balls are green is \(\frac{5}{11}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
When two balls are drawn together, the outcomes are selections of 2 balls out of 12, so the probability of an event is the number of favourable selections divided by the total number of selections, \({}^{12}C_{2}=\frac{12\times11}{2}=66\).
(A) The probability that both balls are red is \(\frac{1}{11}\). Correct. Two of the 4 red balls can be chosen in \({}^{4}C_{2}=6\) ways, so the probability is \(\frac{6}{66}=\frac{1}{11}\).
(B) The probability that one ball is red, and one ball is blue is \(\frac{10}{33}\). Correct. Any one of the 4 red balls can go with any one of the 5 blue balls, giving \(4\times5=20\) selections, so the probability is \(\frac{20}{66}=\frac{10}{33}\).
(C) The probability that both balls are blue is \(\frac{5}{33}\). Correct. Two of the 5 blue balls can be chosen in \({}^{5}C_{2}=10\) ways, so the probability is \(\frac{10}{66}=\frac{5}{33}\).
(D) The probability that both balls are green is \(\frac{5}{11}\). Incorrect. Two of the 3 green balls can be chosen in only \({}^{3}C_{2}=3\) ways, so the probability is \(\frac{3}{66}=\frac{1}{22}\), not \(\frac{5}{11}\).