A bag has 3 red and 2 black balls. Two balls are drawn one after the other without replacement. Which of the following statements are true? I. The probability that both are red is 3/10. II. Given that the first ball is red, the probability that the second is red is 3/5. III. The probability that the second ball is red is 3/5. IV. Red then black and black then red are equally likely.

Question 45

MediumOriginal question

A bag has 3 red and 2 black balls. Two balls are drawn one after the other without replacement. Which of the following statements are true?
I. The probability that both are red is $\frac{3}{10}$ .
II. Given that the first ball is red, the probability that the second is red is $\frac{3}{5}$ .
III. The probability that the second ball is red is $\frac{3}{5}$ .
IV. Red then black and black then red are equally likely.

I, III and IV onlyCorrect answerYour answerI and IV onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerI, II and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

I: $\frac{3}{5} \times \frac{2}{4} = \frac{3}{10}$ . III: $\frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{3}{4} = \frac{3}{5}$ . IV: $\frac{3}{5} \times \frac{2}{4} = \frac{2}{5} \times \frac{3}{4} = \frac{3}{10}$ .
II is false: after a red ball, 2 red balls of 4 remain, so the chance is $\frac{1}{2}$ .

Worth noting

Unseen, the second ball is red with the same chance as the first; once the first is known to be red, the chance falls.

Concept: Probabilities without replacement
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe multiplication theorem

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Ten questions from Conditional Probability & Bayes' Theorem, at your own pace.

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