Question

There are 5 red, 2 yellow and 3 white roses in a flowerpot. One rose is selected from it at random. What is the probability that the selected rose is:
(1) Red
(2) Yellow
(3) Not white colour

Answer 1

Hint: To solve the above question, we will first find out what a probability and its formula are. Then we will find the total number of outcomes if one flower is selected from the total. We will be able to get the total number of outcomes by adding the total number of flowers given in the question. Then, according to the different parts given in the question, we will find the favourable number of outcomes in each part. Then we will apply the formula of probability, which is given by the ratio of favourable outcomes to the total outcomes, in each part to get the answer.

Complete step-by-step answer:
Before, solving the question, we must know what a probability is. The probability of any event is defined as the total chances of happening that event. The probability of any random experiment is denoted by P(E) and it is given by the formula
\[P\left( E \right)=\dfrac{\text{Favorable Outcomes}}{\text{Total number of outcomes}}\]
In our case, the total number of flowers are 5 + 2 + 3 = 10. So, there are a total of 10 roses and we have to select one rose from it. The number of ways in which we can select r things from n things is \[^{n}{{C}_{r}}.\] So, the total number of ways by which we can select one rose from 10 roses is \[^{10}{{C}_{1}}.\] This is also equal to the total number of outcomes. Thus, we can say that,
Total number of outcomes \[={{\text{ }}^{10}}{{C}_{1}}=10\]
Now, we will solve each part given in the question.
(1) Red: If the selected rose is red, then we can select the rose in \[^{5}{{C}_{1}}\] ways. Thus, the favourable outcomes, in this case, are \[^{5}{{C}_{1}}=5.\] Thus the probability becomes
\[P\left( \text{red} \right)=\dfrac{5}{10}=\dfrac{1}{2}\]
(2) Yellow: If the selected rose is yellow, then we can select the rose in \[^{2}{{C}_{1}}\] ways. Thus, the favourable outcomes, in this case, are \[^{2}{{C}_{1}}=2.\] Thus, the probability becomes
\[P\left( \text{yellow} \right)=\dfrac{2}{10}=\dfrac{1}{5}\]
(3) Not white colour: If the selected rose is not white colour, then it will be either red or yellow. Thus, we can select rose in \[\left( ^{5}{{C}_{1}}+{{\text{ }}^{2}}{{C}_{1}} \right)\] ways. Thus, the favourable outcomes become \[^{5}{{C}_{1}}+{{\text{ }}^{2}}{{C}_{1}}=5+2=7.\] Thus, the probability becomes
\[P\left( \text{not white colour} \right)=\dfrac{7}{10}\]

Note: The last part of the question can also be solved in an alternate way which is shown below. The condition in the last part is that the selected rose should not be of white colour. So, first, we will determine the probability when the white rose is selected and then we will subtract it from 1 to get the required probability. Thus,
\[\text{Required Probability}=1-\left( \dfrac{^{3}{{C}_{1}}}{10} \right)\]
\[\Rightarrow \text{Required Probability}=1-\dfrac{3}{10}=\dfrac{7}{10}\]