Question

A purse contains $4$ copper coins, $3$ silver coins, the second purse contains $6$ copper and $2$ silver coins. A coin is taken out of any purse, the probability that it is a copper coin is
(A) $\dfrac{4}{7}$
(B) $\dfrac{3}{4}$
(C) $\dfrac{3}{7}$
(D) $\dfrac{{37}}{{56}}$

Answer 1

Hint : In mathematics probability is a term which is used to indicate the chances of occurrence of an event. Addition theorem of probability is used to solve this problem which state that if $x$ and $y$ are two events then probability is $P\left( {XUB} \right) =$ $P\left( X \right) + P\left( Y \right) - P\left( {X\bigcap Y } \right)$

Complete Step By Step Answer:
As we see in the question there are two purses which contain copper and silver coins.
The chances of choosing any one of the purses is always $50/50$ .
Hence, probability of choosing purse is
Purse $A$ $=$ Purse $B = \dfrac{1}{2}$
As we know purse $A$ has total number of $4$ copper coins and $3$ silver coins, hence, probability of choosing copper coin out of total $7$ coins present in purse will be
$= \dfrac{1}{2} \times \dfrac{4}{7}$
After solving this equation, we get
$= \dfrac{2}{7}$
Hence probability of choosing copper coin from Purse $A$ will be $\left( {\dfrac{2}{7}} \right)$
Similarly, purse $B$ has $6$ copper and $2$ silver coins, therefore probability of choosing copper coin out of total $8$ coins present in purse will be
$= \dfrac{1}{2} \times \dfrac{6}{8}$
After solving this equation, we get
$= \dfrac{3}{8}$
Hence probability of choosing copper coin from Purse $B$ will be $\left( {\dfrac{3}{8}} \right)$
on adding the probability of choosing copper from both the purses will give the final probability of choosing copper from any purse
$= \dfrac{2}{7} + \dfrac{3}{8}$
On taking the L.C.M
$\dfrac{{2 \times 8 + 3 \times 7}}{{56}}$
On solving the above equation, we get
$\dfrac{{16 + 21}}{{56}}$
On further solving we finally get
$\dfrac{{37}}{{56}}$
Hence, the probability of choosing copper coin from any purse will be $\dfrac{{37}}{{56}}$ .Therefore, option $\left( 4 \right)$ is the correct option.

Note :
Value of probability is usually ranging from 0 to 1 where o indicates absence of chances or impossibility of occurrences and 1 indicates certainty. Above condition is defined as a mutually exclusive event which states that two events cannot occur at same time.