Question

A purse contains 4 copper coins and 3 silver coins. A second purse contains 6 copper coins and 4 silver coins. A purse is chosen randomly and a coin is taken out of it. What is the probability that it is a copper coin?
$(a){\text{ }}\dfrac{{41}}{{70}}$
$(b){\text{ }}\dfrac{{31}}{{70}}$
$(c){\text{ }}\dfrac{{27}}{{70}}$
$(d){\text{ }}\dfrac{1}{3}$

Answer 1

Hint: In the above given question, we are asked to find the probability of choosing the copper coin when one of the two purses is chosen randomly. We have to firstly choose one of the two purses and then the copper coin inside the chosen purse. When occurrence of the second event depends on the occurrence of the first event, multiplication is used. When two different events are to be solved together, we add them.

Complete step-by-step answer:
We have the given values as:

The purse ′1′ contains 4 copper coins and 3 silver coins.

The purse ′2′ contains 6 copper coins and 4 silver coins.

Now, we know that the probability of choosing each Purse is$\dfrac{1}{2}$.

Probability of choosing a copper coin = (Probability of choosing a copper coin from Purse 1 after you chose Purse 1) +(Probability of choosing a copper coin from Purse 2 after you chose Purse 1)


= ((Probability of choosing Purse 1)$ \times $(Probability of choosing a copper coin from Purse 1))+((Probability of choosing Purse 2) $ \times $(Probability of choosing a copper coin from Purse 2)) … (1)

Since we are given that, there are 7 coins in purse 1, therefore,

Probability of choosing a copper coin from Purse 1$ = \dfrac{4}{7}$.

Also, there are 10 coins in purse 2, therefore,

Probability of choosing a copper coin from Purse 1$ = \dfrac{6}{{10}}$.


So, after substituting these values in the equation (1), we get,

$ = \left( {\dfrac{1}{2} \times \dfrac{4}{7}} \right) + \left( {\dfrac{1}{2} \times \dfrac{6}{{10}}} \right)$

$ = \dfrac{2}{7} + \dfrac{3}{{10}}$

$ = \dfrac{{10 \times 2 + 3 \times 7}}{{7 \times 10}}$

$ = \dfrac{{20 + 21}}{{70}}$

$ = \dfrac{{41}}{{70}}$

Therefore, the probability of choosing a copper coin from a purse which is chosen randomly is$ = \dfrac{{41}}{{70}}$.

Hence, the solution is option$(a){\text{ }}\dfrac{{41}}{{70}}$.

Note: When we face such a type of problem, we must have a good understanding of probability. We have to calculate the various probabilities with the help of the conditions given and then using these probabilities, we can reach the required solution.