A bag contains an assortment of blue and red balls. If two balls are drawn at random, the probability of drawing two red balls is five times the probability of drawing two blue balls. Furthermore, the probability of drawing one ball of each color is six times the probability of drawing two blue balls. The number of red and blue balls in the bag is :
A.
$$6,\,3$$
B.
$$3,\,6$$
C.
$$2,\,7$$
D.
none of these
Answer :
$$6,\,3$$
Solution :
Let the number of red and blue balls be $$r$$ and $$b$$, respectively. Then, the probability of drawing two red balls
$$ = {p_1} = \frac{{{}^r{C_2}}}{{{}^{r + b}{C_2}}} = \frac{{r\left( {r - 1} \right)}}{{\left( {r + b} \right)\left( {r + b - 1} \right)}}$$
The probability of drawing two blue balls is
$$ = {p_2} = \frac{{{}^b{C_2}}}{{{}^{r + b}{C_2}}} = \frac{{b\left( {b - 1} \right)}}{{\left( {r + b} \right)\left( {r + b - 1} \right)}}$$
The probability of drawing one red and one blue ball
$$ = {p_3} = \frac{{{}^r{C_1}{}^b{C_1}}}{{{}^{r + b}{C_2}}} = \frac{{2br}}{{\left( {r + b} \right)\left( {r + b - 1} \right)}}$$
By hypothesis, $${p_1} = 5{p_2}$$ and $${p_3} = 6{p_2}.$$
Therefore, $$r\left( {r - 1} \right) = 5b\left( {b - 1} \right)$$ and $$2br = 6b\left( {b - 1} \right) \Rightarrow r = 6,\,b = 3.$$
Releted MCQ Question on Statistics and Probability >> Probability
Releted Question 1
Two fair dice are tossed. Let $$x$$ be the event that the first die shows an even number and $$y$$ be the event that the second die shows an odd number. The two events $$x$$ and $$y$$ are:
Two events $$A$$ and $$B$$ have probabilities 0.25 and 0.50 respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is 0.14. Then the probability that neither $$A$$ nor $$B$$ occurs is
The probability that an event $$A$$ happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. The probability that the event $$A$$ happens at least once is
If $$A$$ and $$B$$ are two events such that $$P(A) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {\frac{{\overline A }}{{\overline B }}} \right)$$ is equal to
(Here $$\overline A$$ and $$\overline B$$ are complements of $$A$$ and $$B$$ respectively).
A.
$$1 - P\left( {\frac{A}{B}} \right)$$
B.
$$1 - P\left( {\frac{{\overline A }}{B}} \right)$$
C.
$$\frac{{1 - P\left( {A \cup B} \right)}}{{P\left( {\overline B } \right)}}$$
D.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$