Question

The probability that the three cards are drawn from a pack of \[52\] cards are all red is
(1) \[\dfrac{1}{17}\]
(2) \[\dfrac{3}{19}\]
(3) \[\dfrac{2}{19}\]
(4) \[\dfrac{2}{17}\]

Answer 1

Hint: In this question, first we will calculate the total number of outcomes and that will be calculated with the help of a combination. Then we will find the total number of favorable outcomes that will also be calculated with the help of combination and after that, we will calculate the probability by dividing the favorable outcome by the total number of the outcome.

Complete step-by-step answer:
Using probability we can find the occurrence of an event. By probability, we mean the possibility of an event. The probability of an event will always come to less than zero. If the probability of an event comes out to zero then that event is considered an impossible event. The formula of probability can be expressed as the ratio of the number of favorable outcomes with the total number of outcomes. The sum of all possible outcomes in an event is always equal to one.
In a group of \[52\] cards, we have \[26\] red cards and \[26\] black cards. The \[26\] black cards are further divided into \[13\] spades and \[13\] clubs. The \[26\] red cards are also divided into \[13\] hearts and \[13\] diamonds. The pack of \[13\] cards contains one king, one queen, one ace, one jack, and all the numbers from two to ten.
 In the above question, we have to find the probability of the three red cards drawn from the pack of \[52\] cards.
So the total number of outcomes will be \[{}^{52}{{C}_{3}}\] because the total number of cards are \[52\] and the number of a favorable outcome will be \[{}^{26}{{C}_{3}}\] because we have to choose three red cards and the total number of red cards are \[26\] . Now the formula of probability will be given as shown below.
 \[probability=\dfrac{\text{Number of favourable outcomes}}{\text{Number of total outcomes}}\]
So probability will be
 \[probability=\dfrac{{}^{26}{{C}_{3}}}{{}^{52}{{C}_{3}}}\]
The above equation will be solved with the help of a combination. So the result will be as follows
 \[probability=\dfrac{\dfrac{\left| \!{\underline {\,
  26 \,}} \right. }{\left| \!{\underline {\,
  3 \,}} \right. \left| \!{\underline {\,
  26-3 \,}} \right. }}{\dfrac{\left| \!{\underline {\,
  52 \,}} \right. }{\left| \!{\underline {\,
  3 \,}} \right. \left| \!{\underline {\,
  52-3 \,}} \right. }}\]

 \[\Rightarrow probability=\dfrac{\dfrac{26\times 25\times 24\times \left| \!{\underline {\,
  23 \,}} \right. }{\left| \!{\underline {\,
  23 \,}} \right. }}{\dfrac{52\times 51\times 50\times \left| \!{\underline {\,
  49 \,}} \right. }{\left| \!{\underline {\,
  49 \,}} \right. }}\]

 \[\Rightarrow probability=\dfrac{26\times 25\times 24}{52\times 51\times 50}\]
 \[\Rightarrow probability=\dfrac{2}{17}\]
So the probability of choosing three red cards from the pack of \[52\] cards will be \[\dfrac{2}{17}\] .
So, the correct answer is “Option 4”.

Note: The sample space in a probability is defined as the set of all possible outcomes that can happen in an event. If there are two events such that the occurrence of one event does not depend upon the occurrence of another event then both the events are known as independent events.