Question

What is the probability of drawing a 6 from a deck of cards?

Answer 1

Consider that there are four suits of cards namely: spades, clubs, diamonds and hearts and each suit has 13 cards in them. Now, find the total number of cards which will behave as the total number of sample space $n\left( S \right)$. Now, considering the fact that each suit has a card 6, find the total number of favorable outcomes $n\left( E \right)$. Use the formula of probability given as $P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$ to get the answer.

Complete step-by-step solution:
Here we have asked to determine the probability of drawing a 6 from a deck of cards. First we need to find the total number of cards in a deck and the total numbers of card 6 present in them.
Now, we know that a pack of card contains 4 suits of different cards namely: spades, clubs, diamonds and hearts and each suit has 13 cards in them. Therefore the total number of card in a deck is given as: -
$\begin{aligned} & \Rightarrow n\left( S \right)=13+13+13+13 \\\ & \Rightarrow n\left( S \right)=52 \\\ \end{aligned}$
Here, n (S) is the total number of sample space.
Now, a suit of 13 cards has a king, a queen, a jack, an ace and 9 cards numbered from 2 to 9. This means that in each suit there will be card numbered 6, so in four suits we will have four cards number 6. So we have,
$\Rightarrow n\left( E \right)=4$
Here, n (E) is the total number of favourable outcomes.
Now, the probability of an event is the ratio of total number of favourable outcomes to the total number of outcomes so we get the probability P (E) of drawing a 6 from the deck of cards as: -
$\begin{aligned} & \Rightarrow P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)} \\\ & \Rightarrow P\left( E \right)=\dfrac{4}{52} \\\ & \therefore P\left( E \right)=\dfrac{1}{13} \\\ \end{aligned}$
Hence, the above obtained ratio is our answer.

Note: You must remember the total number of cards present in a pack or deck of cards. Note that in the four suits of different cards we have two suits black and two suits red in color. Spades and clubs are black while hearts and diamonds are red. Remember the formula of probability of an event to occur to solve the above question.