Question

A bag contains $4$ white and $3$ red balls. Two draws of one ball each are made without replacement. Then the probability that both the balls are red is
A) $\dfrac{1}{7}$
B) $\dfrac{2}{7}$
C) $\dfrac{3}{7}$
D) $\dfrac{4}{7}$

Answer 1

Hint: Given, that a bag contains white and red balls. Firstly, we will find the total number of balls. Then, we find the required probability with the help of a combination formula and it is also given that two draws of one ball each are made without replacement.

Formula Used:
${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$

Complete step by step Solution:
Given, that a bag contains $4$ white and $3$red balls.
Total number of balls$ = 4 + 3$
$ = 7$
Balls are drawn without replacement
Hence, the probability that both the balls are red is $ = \dfrac{{{}^3{C_1}}}{{{}^7{C_1}}}.\dfrac{{{}^2{C_1}}}{{{}^6{C_1}}}$
We know ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
On solving
$ = \dfrac{{\dfrac{{3!}}{{2!}}}}{{\dfrac{{7!}}{{6!}}}}.\dfrac{{\dfrac{{2!}}{{1!}}}}{{\dfrac{{6!}}{{5!}}}}$
Simplifying
$ = \dfrac{3}{7}.\dfrac{2}{6}$
$ = \dfrac{1}{7}$

Hence, the correct option is (A).

Additional Information:
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The probability is between zero and one. To predict how likely events are to happen probability has been introduced in mathematics. The meaning of probability is basically the extent to which something is likely to happen.
Probability without replacement involves dependent events where the preceding event has an effect on the probability of the next event. Probability without replacement means once we draw an item, then we do not replace it back to the sample space before drawing a second item. In other words, an item cannot be drawn more than once. If the object is put back in the group before an object is chosen again, we call it sampling with replacement. If the object is put to one side, we call it sampling without replacement.

Note: Students can make mistakes while finding probability out of only red balls. They should find probability from the total number of balls. And take care of the fact that balls are drawn without replacement to find an accurate answer.