Question

What is the chance of drawing a sovereign from a purse one compartment of which contains 3 shillings and 2 sovereigns, and the other 2 sovereigns and 1 shilling?

Answer 1

Hint: Identify the probabilities of each compartment. Calculate the conditional probabilities by counting the number of total items. Use the total probability theorem after obtaining all these values.

Complete step-by-step answer:
Let us divide the compartments as A and B.
Compartment A has 3 shillings and 2 Sovereigns.
Compartment B has 1 shilling and 2 Sovereigns.

| A| B
---|---|---
Shillings| 3| 1
Sovereigns| 2| 2

Since there are two compartments, probability of choosing any one is $\dfrac{1}{2}$
$\Rightarrow {\text{P}}\left( {\text{A}} \right) = {\text{P}}\left( {\text{B}} \right) = \dfrac{1}{2}$
Total number of objects in compartment A=5
Number of sovereigns = 2

Hence the probability of choosing a sovereign = $\dfrac{{{\text{number of sovereigns}}}}{{{\text{total number of objects}}}}$
$\Rightarrow {\text{P}}\left( {\dfrac{{{\text{SO}}}}{{\text{A}}}} \right) = \dfrac{2}{5}$.
Total number of objects in compartment B=3
Number of sovereigns = 2

Hence the probability of choosing a sovereign = $\dfrac{{{\text{number of sovereigns}}}}{{{\text{total number of objects}}}}$
$\Rightarrow {\text{P}}\left( {\dfrac{{{\text{SO}}}}{{\text{B}}}} \right) = \dfrac{2}{3}$.

Now we make use of total probability theorem, which is
For probability of sovereigns ${\text{P}}\left( {{\text{SO}}} \right) = {\text{P}}\left( {\text{A}} \right){\text{P}}\left( {\dfrac{{{\text{SO}}}}{{\text{A}}}} \right) + {\text{P}}\left( {\text{B}} \right){\text{P}}\left( {\dfrac{{{\text{SO}}}}{{\text{B}}}} \right) \\\ \Rightarrow {\text{P}}\left( {{\text{SO}}} \right) = \left( {\dfrac{1}{2} \times \dfrac{2}{5}} \right){\text{ + }}\left( {\dfrac{1}{2} \times \dfrac{2}{3}} \right) \\\ \Rightarrow {\text{P}}\left( {{\text{SO}}} \right) = \dfrac{8}{{15}} \\\$

Note: In such types of questions, identifying the conditional probabilities and knowing how to use the total probability theorem is the key. Conditional probability is the measure of probability of one event with respect to another even or knowing that the other event has already occurred.