Question

# One purse contains $1$ sovereign and $3$ shillings, a second purse contains $2$ sovereigns and $4$ shillings, and a third 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.

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Hint: Here we find chances of a coin being a sovereign from all three purses separately
* If there are $n$ equally likely objects then the chance of selecting one object from $n$ equally likely objects is equal to $\dfrac{1}{n}$.
* Rule of product: If there are $m$ ways to do something and $n$ to do another thing, then there are $m \times n$ ways to do both things.
* Rule of sum: if there are ${r_1}$ possible outcomes for an event and ${r_2}$ possible outcomes for another event and the two events cannot both occur, then there are ${r_1} + {r_2}$ total possible outcomes for the events.

Since each purse is equally likely to be taken, the change of selection one purse from total $3$ purses is $\dfrac{1}{3}.$
Find the chance of drawing a sovereign from the first purse.
There are $4$ ($1$ sovereign$+ 3$ Shillings) coins in first purse, hence the chance of drawing a sovereign is given by number of shillings divided by number of coins $= \dfrac{1}{4}$
$\therefore$ Chance of drawing a sovereign from first purse is $\dfrac{1}{3} \times \dfrac{1}{4} = \dfrac{1}{{12}}$ (Using rule of product)
Find the chance of drawing a sovereign from the second purse.
There are $6$ ($2$ sovereign$+ 4$ Shillings) coins in second purse, hence the chance of drawing a sovereign is given by number of shillings divided by number of coins $= \dfrac{2}{6}$
$\therefore$ Chance of drawing a sovereign from second purse is $\dfrac{1}{3} \times \dfrac{2}{6} = \dfrac{1}{9}$ (Using rule of product)
Find the chance of drawing a sovereign from the third purse.
There are $4$ ($3$ sovereign$+ 1$ Shillings) coins in third purse, hence the chance of drawing a sovereign is given by number of shillings divided by number of coins $= \dfrac{3}{4}$
$\therefore$ Chance of drawing a sovereign from second purse is $\dfrac{1}{3} \times \dfrac{3}{4} = \dfrac{1}{4}$ (Using rule of product)
Find a chance of getting a sovereign, when a coin is taken out of one of the purses randomly. We calculate the sum of all chances from the first, second and third purse.
Therefore, using the rule of sum, where each event of taking out a sovereign from each purse is an independent event.
$\therefore$ Chance of coin to be a sovereign $= \dfrac{1}{{12}} + \dfrac{1}{9} + \dfrac{1}{4}$
By taking LCM on the RHS of the equation.
$= \dfrac{{3 + 4 + 9}}{{36}} \\ = \dfrac{{16}}{{36}} \\ = \dfrac{4}{9} \\$
Thus, the chance of a coin to be a sovereign is $\dfrac{4}{9}$.

Note: Students are likely to make mistakes applying combination or permutation formulas to this question which is wrong because here we don’t have to find a number of ways in which we can pick a sovereign, we have to find a chance or in other words probability.