Question

A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled at random from one of these two purses, what is the probability that it is a silver coin?

Answer 1

Consider the probability of selecting the first or the second purse before calculating the probabilities of picking out a silver coin from either one of them. The probability of picking out a silver coin from a purse will be the product of the probability of picking that purse and the probability of picking a silver coin, given that that particular purse is already selected.
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Complete step by step answer: -**
To tackle this question, we first need to understand all the possible cases in which we might get a silver coin.
It might be that we select a coin from the first purse, and the coin is a silver coin, or it might be that we select a coin from the second purse, and the coin is a silver coin then. Either way, we also need to account for the probability of selecting the first or the second purse.
Now, for the first purse, the probability of choosing a coin from the first purse will be = $P(Purse1)=\dfrac{1}{2}$, since the purses are unbiased and there’s equal probability of me choosing either of the two purses. Similarly, probability of choosing the second purse = $P(Purse2)=\dfrac{1}{2}$.
Now, for purse 1, the probability of selecting a silver coin = $\dfrac{silver}{total}$= the total number of silver coins in the purse divided by the total number of coins that are there in purse 1.
No. of silver coins in purse 1 = 2
Total no. of coins in purse 1 = 6
Therefore, probability of taking out a silver coin, given that we have selected purse 1 already = $P(silver1)=\dfrac{2}{6}=\dfrac{1}{3}$.
Now, coming to the second purse, similarly, the probability of choosing a silver coin from the second purse = $\dfrac{silver}{total}$= total number of silver coins in purse 2 divided by the total number of coins in purse 2.
No. of silver coins in purse 2 = 4
Total no. of coins in purse 2 = 7

Therefore, the probability of taking out a silver coin, given that we have selected purse 2 already = $P(silver2)=\dfrac{4}{7}$.
Now, we need to find the probability of choosing a silver coin. We need to take in consideration the probability of selecting purse 1 and purse 2 respectively.
Therefore, probability of selecting a silver coin = $\begin{aligned} & P(silver)=P(Purse1).P(silver1)+P(Purse2).P(silver2) \\\ & =\dfrac{1}{2}.\dfrac{1}{3}+\dfrac{1}{2}.\dfrac{4}{7}=\dfrac{1}{2}(\dfrac{1}{3}+\dfrac{1}{7})=\dfrac{1}{2}\times \dfrac{10}{21}=\dfrac{5}{21} \\\ \end{aligned}$
Hence, we have the probability of selecting a silver coin = $\dfrac{5}{21}$.
The required probability = $\dfrac{5}{21}$.

Note: Don’t forget to consider the probabilities of selecting purse 1 or purse 2. Often, students forget to do that and end up with the wrong answer. You might think that we can just find the probability by dividing the number of silver coins by the total number of coins in both the purses. But, we can’t do that. An easy way to think would be to analyse the number of steps the person who actually is choosing has to go through. He’ll first pick the purse, then pick a coin, and thus, we need to consider possibilities at each and every step.