Question

A bag contains some green, white and pink beads. The probability of taking out one green bead is $\dfrac{1}{3}$ and that of picking up one pink bead is$\dfrac{1}{4}$. If it is known that the box has 10 white beads then how many beads were in the box initially?
A.24
B.25
C.28
D.32

Answer 1

Hint:
Probability is a type of ratio where we compare how many times an outcome can occur compared to all possible outcomes.
${\text{Probability = }}\dfrac{{{\text{Number of required outcomes}}}}{{{\text{Number of possible outcomes}}}}$
Independent events: Two events are independent when the outcomes of the first event do not influence the outcome of the second event. To find the probability of two independent events:
${\text{P(X and Y) = P(X)}}{\text{.P(Y)}}$
Dependent events: Two events are dependent when the outcome of the first event influences the outcome of the second event. To find the probability of two dependent events:
${\text{P(X and Y) = P(X)}}{\text{.P(Y after X)}}$
Mutually exclusive events: Two events are mutually exclusive when two events cannot happen at the same time. To find the probability of two mutually exclusive events:
${\text{P(X or Y) = P(X) + P(Y)}}$
Let green, white and pink beads be a, b and c. Now apply the conditions which are given in the questions.

Complete step by step solution:
Let green, white and pink beads be a, b and c respectively.
Given, probability of taking out one green bead is $\dfrac{1}{3}$
\[\therefore \dfrac{a}{{a + b + c}} = \dfrac{1}{3}\]
\[ \Rightarrow 3a = a + b + c\]
$ \Rightarrow 2a = b + c$$ \ldots \left( 1 \right)$
Now, again probability of taking one pink bead is$\dfrac{1}{4}$
 \[\begin{gathered}
  \therefore \dfrac{c}{{a + b + c}} = \dfrac{1}{4} \\
   \Rightarrow 4c = a + b + c \\
   \Rightarrow 3c = a + b \ldots \left( 2 \right) \\
\end{gathered} \]
Again, given that basket has 10 green beads.
$\therefore b = 10 \ldots \left( 3 \right)$
Putting the value of $b = 10$ in equation (1) & (2) we get
$2a = 10 + c \ldots \left( 4 \right)$
$3c = a + 10 \ldots \left( 5 \right)$
From equation (5) putting $a = 3c - 10$ in equation (4) we get
$2\left( {3c - 10} \right) = 10 + c$
$6c - 20 = 10 + c$
$5c = 30$
$ \Rightarrow c = 6$
Putting the value of c in equation (5)
$ \Rightarrow 3 \times 6 = a + 10$
$ \Rightarrow a = 18 - 10$
$ \Rightarrow a = 8$
$\therefore {\text{Total beads in the basket }} = {\text{ }}a + b + c$
$ = 8 + 10 + 6$
$ = 24{\text{ beads}}$

Note: Another way of representing 2 or more events is on a probability tree. It follows the method given below:
The probability of each branch is written on the branch.
The outcome is written at the end of the branch.
We multiply probabilities along the branches.
We add probabilities down columns.
All probabilities add to 1.