# There are three boys and two girls. A committee of two is to be formed. Find the probability of the events that the committee contains at least one girl.

Last updated date: 19th Mar 2023

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Hint: We have to find the probability of the events that the committee contains at least one girl. So consider a sample space $(S)$ and write the number of committees that can be formed in sample space$n(S)$. Then find a set $A$ that contains at least one girl in the committee and $n(A)$. Use $P(A)=\dfrac{n(A)}{n(S)}$, you will get the answer.

Complete step-by-step answer:

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. Probability is a number between $0$ and $1$, where, roughly speaking, $0$ indicates impossibility and $1$ indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is $\dfrac{1}{2}$ (which could also be written as $0.5$ or$50%$).

The theory of probability is a representation of its concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.

Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results.

So in the given question, there are three boys and two girls.

Let the boys are ${{B}_{1}},$ ${{B}_{2}},$ ${{B}_{3}},$and girls are ${{G}_{1}},$ ${{G}_{2}}$.

It is given that a committee of two is formed.

Thus the sample space$(S)$ becomes,

$S=\left\{ {{B}_{1}}{{B}_{2}},{{B}_{1}}{{B}_{3}},{{B}_{2}}{{B}_{3}},{{B}_{1}}{{G}_{1}},{{B}_{1}}{{G}_{2}},{{B}_{2}}{{G}_{1}},{{B}_{2}}{{G}_{2}},{{B}_{3}}{{G}_{1}},{{B}_{3}}{{G}_{2}},{{G}_{1}}{{G}_{2}} \right\}$

Here $n(S)=10$.

Let $A$ be the event that the committee contains at least one girl.

Then $A=\left\{ {{B}_{1}}{{G}_{1}},{{B}_{1}}{{G}_{2}},{{B}_{2}}{{G}_{1}},{{B}_{2}}{{G}_{2}},{{B}_{3}}{{G}_{1}},{{B}_{3}}{{G}_{2}},{{G}_{1}}{{G}_{2}} \right\}$

Here $n(A)=7$.

Probability of the events that the committee contain at least one girl $=P(A)=\dfrac{n(A)}{n(S)}$

So $P(A)=\dfrac{7}{10}$.

So we get the final answer as the probability of the events that the committee contains at least one girl is $\dfrac{7}{10}$.

Note: Carefully read the question. You must know what sample space is. Do not confuse yourself. You should know the probability $P(A)=\dfrac{n(A)}{n(S)}$. You should be also familiar with the events such as we have an event $A$ that contains a committee of at least one girl.

Complete step-by-step answer:

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. Probability is a number between $0$ and $1$, where, roughly speaking, $0$ indicates impossibility and $1$ indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is $\dfrac{1}{2}$ (which could also be written as $0.5$ or$50%$).

The theory of probability is a representation of its concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.

Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results.

So in the given question, there are three boys and two girls.

Let the boys are ${{B}_{1}},$ ${{B}_{2}},$ ${{B}_{3}},$and girls are ${{G}_{1}},$ ${{G}_{2}}$.

It is given that a committee of two is formed.

Thus the sample space$(S)$ becomes,

$S=\left\{ {{B}_{1}}{{B}_{2}},{{B}_{1}}{{B}_{3}},{{B}_{2}}{{B}_{3}},{{B}_{1}}{{G}_{1}},{{B}_{1}}{{G}_{2}},{{B}_{2}}{{G}_{1}},{{B}_{2}}{{G}_{2}},{{B}_{3}}{{G}_{1}},{{B}_{3}}{{G}_{2}},{{G}_{1}}{{G}_{2}} \right\}$

Here $n(S)=10$.

Let $A$ be the event that the committee contains at least one girl.

Then $A=\left\{ {{B}_{1}}{{G}_{1}},{{B}_{1}}{{G}_{2}},{{B}_{2}}{{G}_{1}},{{B}_{2}}{{G}_{2}},{{B}_{3}}{{G}_{1}},{{B}_{3}}{{G}_{2}},{{G}_{1}}{{G}_{2}} \right\}$

Here $n(A)=7$.

Probability of the events that the committee contain at least one girl $=P(A)=\dfrac{n(A)}{n(S)}$

So $P(A)=\dfrac{7}{10}$.

So we get the final answer as the probability of the events that the committee contains at least one girl is $\dfrac{7}{10}$.

Note: Carefully read the question. You must know what sample space is. Do not confuse yourself. You should know the probability $P(A)=\dfrac{n(A)}{n(S)}$. You should be also familiar with the events such as we have an event $A$ that contains a committee of at least one girl.

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