# A class consists of \[100\] students, \[25\] of them are girls and \[75\] are boys, \[20\] of them are rich and remaining poor, \[40\] of them are fair complexioned. The probability of selecting a fair complexioned rich girl is

.

Answer

Verified

362.7k+ views

Hint: Calculate the probability of each of the given events and use the formula for calculating the probability of intersection of independent events, which states that the probability of intersection of independent events is the product of probability of each of the individual events.

Complete step-by-step answer:

We have the data regarding a class of \[100\] students. We have to find the probability of selecting a fair complexioned rich girl.

Let’s denote the event of getting a girl by \[A\] and thus, the event of getting a boy is denoted by \[{{A}^{c}}\].

Similarly, denote the event of getting a rich person by \[B\] and thus, the event of getting a poor person is denoted by \[{{B}^{c}}\].

Similarly, the event of getting a fair complexioned person is denoted by \[C\] and thus, the event of getting a dark complexioned person is denoted by \[{{C}^{c}}\].

We have to calculate the probability of each of these events. We know that the probability of getting any event is the ratio of the number of favourable outcomes to the total number of possible outcomes.

As there are \[25\] girls in a class, we have \[P\left( A \right)=\dfrac{25}{100}=\dfrac{1}{4}\] and \[P\left( {{A}^{c}} \right)=\dfrac{75}{100}=\dfrac{3}{4}\].

Similarly, as we have \[20\] rich students in a class, we have \[P\left( B \right)=\dfrac{20}{100}=\dfrac{1}{5}\] and \[P\left( {{B}^{c}} \right)=\dfrac{80}{100}=\dfrac{4}{5}\].

As there are \[40\] fair complexioned students in a class, we have \[P\left( C \right)=\dfrac{40}{100}=\dfrac{2}{5}\] and \[P\left( {{C}^{c}} \right)=\dfrac{60}{100}=\dfrac{3}{5}\].

We now have to evaluate the probability of getting a fair complexioned rich girl, which is \[P\left( A\cap B\cap C \right)\]. We observe that the events \[A,B,C\] are independent, which means that the occurrence or non-occurrence of one event doesn’t affect the occurrence or non-occurrence of another event.

We know that if events \[A,B,C\] are independent, we have \[P\left( A\cap B\cap C \right)=P\left( A \right)P\left( B \right)P\left( C \right)\].

Substituting the values from above equations, we have \[P\left( A\cap B\cap C \right)=P\left( A \right)P\left( B \right)P\left( C \right)=\dfrac{1}{4}\times \dfrac{1}{5}\times \dfrac{2}{5}=\dfrac{1}{50}\].

Hence, the probability of getting a fair complexioned rich girl, denoted by \[P\left( A\cap B\cap C \right)\], is equal to \[\dfrac{1}{50}\].

Note: Probability of any event describes how likely an event is to occur or how likely it is that a proposition is true. The value of probability of any event always lies in the range \[\left[ 0,1 \right]\] where having \[0\] probability indicates that the event is impossible to happen, while having probability equal to \[1\] indicates that the event will surely happen. We must remember that the sum of probability of occurrence of some event and probability of non-occurrence of the same event is always \[1\].

Complete step-by-step answer:

We have the data regarding a class of \[100\] students. We have to find the probability of selecting a fair complexioned rich girl.

Let’s denote the event of getting a girl by \[A\] and thus, the event of getting a boy is denoted by \[{{A}^{c}}\].

Similarly, denote the event of getting a rich person by \[B\] and thus, the event of getting a poor person is denoted by \[{{B}^{c}}\].

Similarly, the event of getting a fair complexioned person is denoted by \[C\] and thus, the event of getting a dark complexioned person is denoted by \[{{C}^{c}}\].

We have to calculate the probability of each of these events. We know that the probability of getting any event is the ratio of the number of favourable outcomes to the total number of possible outcomes.

As there are \[25\] girls in a class, we have \[P\left( A \right)=\dfrac{25}{100}=\dfrac{1}{4}\] and \[P\left( {{A}^{c}} \right)=\dfrac{75}{100}=\dfrac{3}{4}\].

Similarly, as we have \[20\] rich students in a class, we have \[P\left( B \right)=\dfrac{20}{100}=\dfrac{1}{5}\] and \[P\left( {{B}^{c}} \right)=\dfrac{80}{100}=\dfrac{4}{5}\].

As there are \[40\] fair complexioned students in a class, we have \[P\left( C \right)=\dfrac{40}{100}=\dfrac{2}{5}\] and \[P\left( {{C}^{c}} \right)=\dfrac{60}{100}=\dfrac{3}{5}\].

We now have to evaluate the probability of getting a fair complexioned rich girl, which is \[P\left( A\cap B\cap C \right)\]. We observe that the events \[A,B,C\] are independent, which means that the occurrence or non-occurrence of one event doesn’t affect the occurrence or non-occurrence of another event.

We know that if events \[A,B,C\] are independent, we have \[P\left( A\cap B\cap C \right)=P\left( A \right)P\left( B \right)P\left( C \right)\].

Substituting the values from above equations, we have \[P\left( A\cap B\cap C \right)=P\left( A \right)P\left( B \right)P\left( C \right)=\dfrac{1}{4}\times \dfrac{1}{5}\times \dfrac{2}{5}=\dfrac{1}{50}\].

Hence, the probability of getting a fair complexioned rich girl, denoted by \[P\left( A\cap B\cap C \right)\], is equal to \[\dfrac{1}{50}\].

Note: Probability of any event describes how likely an event is to occur or how likely it is that a proposition is true. The value of probability of any event always lies in the range \[\left[ 0,1 \right]\] where having \[0\] probability indicates that the event is impossible to happen, while having probability equal to \[1\] indicates that the event will surely happen. We must remember that the sum of probability of occurrence of some event and probability of non-occurrence of the same event is always \[1\].

Last updated date: 02nd Oct 2023

•

Total views: 362.7k

•

Views today: 4.62k

Recently Updated Pages

What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is meant by shramdaan AVoluntary contribution class 11 social science CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

An alternating current can be produced by A a transformer class 12 physics CBSE

What is the value of 01+23+45+67++1617+1819+20 class 11 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers