Question

One purse contains 1 sovereign and 3 shillings, a second purse contains 2 sovereigns and 4 shillings, and a third purse contains 3 sovereigns and 1 shilling. If a coin is taken out of one of the purses selected at random, find the chance that it is a sovereign.

Answer 1

In this question for finding the probability of getting a sovereign we need to find the probability of getting a sovereign in each purse because of the events of getting a sovereign from three different purses are independent events. This is the combination of three independent events. So, for finding the probability of getting a sovereign we find the probability of getting sovereign in each event and add them all.

Complete step-by-step solution
Let us assume that $'E'$ be the event of getting sovereign from the collection of three purses.
Let us assume $'{{E}_{1}}’, ‘{{E}_{2}}’, ‘{{E}_{3}}'$ are the events of getting sovereign from purse 1, 2, 3 respectively.
Now, let us find the probability of event ${{E}_{1}}$ that is the probability of getting sovereign from purse 1
Here, we know that there are three purses, so the probability of getting the first purse is
$\Rightarrow P\left( \text{getting purse 1} \right)=\dfrac{1}{3}$
We are given that purse 1 has 1 sovereign and 3 shillings.
So, the probability of getting sovereign from purse 1 is
$\Rightarrow P\left( \text{getting sovereign from purse 1} \right)=\dfrac{1}{4}$
Now, the probability of event ${{E}_{1}}$ is given as

& \Rightarrow P\left( {{E}_{1}} \right)=P\left( \text{getting purse 1} \right)\times P\left( \text{getting sovereign from purse 1} \right) \\\ & \Rightarrow P\left( {{E}_{1}} \right)=\dfrac{1}{3}\times \dfrac{1}{4}=\dfrac{1}{12} \\\ \end{aligned}$$ Now, let us find probability of event $${{E}_{2}}$$ that is probability of getting sovereign from purse 2 Here, we know that there are three purses, so the probability of getting second purse is $$\Rightarrow P\left( \text{getting purse 2} \right)=\dfrac{1}{3}$$ We are given that purse 2 has 2 sovereign and 4 shillings. So, the probability of getting sovereign from purse 2 is $$\Rightarrow P\left( \text{getting sovereign from purse 2} \right)=\dfrac{2}{6}$$ Now, the probability of event $${{E}_{2}}$$ is given as $$\begin{aligned} & \Rightarrow P\left( {{E}_{2}} \right)=P\left( \text{getting purse 2} \right)\times P\left( \text{getting sovereign from purse 2} \right) \\\ & \Rightarrow P\left( {{E}_{2}} \right)=\dfrac{1}{3}\times \dfrac{2}{6}=\dfrac{1}{9} \\\ \end{aligned}$$ Now, let us find the probability of event $${{E}_{3}}$$ that is the probability of getting sovereign from purse 3 Here, we know that there are three purses, so the probability of getting the third purse is $$\Rightarrow P\left( \text{getting purse 3} \right)=\dfrac{1}{3}$$ We are given that purse 1 has 3 sovereign and 1 shilling. So, the probability of getting sovereign from purse 3 is $$\Rightarrow P\left( \text{getting sovereign from purse 3} \right)=\dfrac{3}{4}$$ Now, the probability of event $${{E}_{3}}$$ is given as $$\begin{aligned} & \Rightarrow P\left( {{E}_{3}} \right)=P\left( \text{getting purse 3} \right)\times P\left( \text{getting sovereign from purse 3} \right) \\\ & \Rightarrow P\left( {{E}_{3}} \right)=\dfrac{1}{3}\times \dfrac{3}{4}=\dfrac{1}{4} \\\ \end{aligned}$$ We know that the probability of getting sovereign from set of three purses is given as the combination of probabilities of $$'{{E}_{1}}’, ’{{E}_{2}}, ‘{{E}_{3}}'$$ that is $$\Rightarrow P\left( E \right)=P\left( {{E}_{1}} \right)+P\left( {{E}_{2}} \right)+P\left( {{E}_{3}} \right)$$ Now, by substituting the values of probabilities in above equation we get $$\begin{aligned} & \Rightarrow P\left( E \right)=\dfrac{1}{12}+\dfrac{1}{9}+\dfrac{1}{4} \\\ & \Rightarrow P\left( E \right)=\dfrac{3+4+9}{36} \\\ & \Rightarrow P\left( E \right)=\dfrac{16}{36}=\dfrac{4}{9} \\\ \end{aligned}$$ **Therefore, the probability of getting a sovereign when a coin is taken at random from the given purses is $$\dfrac{4}{9}$$.** **Note:** Students will make mistakes in considering the question as a combination question. As it was a combination of independent events they consider that there is one purse having a total of 6 sovereign and 8 shillings. They combine before calculating the probability. We need to apply combinations to probabilities, not for coins. The remaining calculations need to be taken care of.