A box contains 5 blue balls, 3 black balls and 2 red balls of the same size. A ball is selected at random from the box and then replaced. A second ball is t...
Question 1 Report
A box contains 5 blue balls, 3 black balls and 2 red balls of the same size. A ball is selected at random from the box and then replaced. A second ball is then selected. Find the probability of obtaining
(a) two red balls ;
(b) two blue balls or two black balls ;
(c) one black and one red ball in any order.
Answer Details
Total balls \(= 5 + 3 + 2 = 10\). Since each ball is replaced, the two draws are independent and the probabilities on each draw stay the same.
\(P(\text{blue}) = \tfrac{5}{10} = \tfrac{1}{2},\quad P(\text{black}) = \tfrac{3}{10},\quad P(\text{red}) = \tfrac{2}{10} = \tfrac{1}{5}\)
(a) Two red balls:
\(P = \tfrac{1}{5}\times\tfrac{1}{5} = \dfrac{1}{25}\)
(b) Two blue OR two black (mutually exclusive, so add):
\(P = \left(\tfrac{1}{2}\right)^2 + \left(\tfrac{3}{10}\right)^2 = \tfrac{1}{4} + \tfrac{9}{100} = \tfrac{25}{100} + \tfrac{9}{100} = \dfrac{34}{100} = \dfrac{17}{50}\)
(c) One black and one red in any order (black-then-red or red-then-black):
\(P = 2\times\tfrac{3}{10}\times\tfrac{2}{10} = 2\times\tfrac{6}{100} = \dfrac{12}{100} = \dfrac{3}{25}\)
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