Question

From a pack of playing cards all cards whose numbers are multiples of 3 are removed. A card is drawn at random from remaining cards. Then the probability that the card is drawn is an even number which is red card is:
(a) \[\dfrac{10}{52}\]
(b) \[\dfrac{1}{4}\]
(c) \[\dfrac{1}{5}\]
(d) \[\dfrac{3}{13}\]

Answer 1

Hint: We solve this problem first by removing all the 3 multiple cards from the deck of playing cards.
In a deck of 52 cards there will be 4 sets of 13 cards each in which two are black and two are red. Each set has 13 cards named numbers 2 to 10 and one ace, king, queen and jack.
By using this information we find the remaining cards after removing all the 3 multiple cards then we find the required probability by using the formula that is the formula of probability is given as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]

Complete step by step answer:
We are given that there is a deck of playing cards.
We know that there will be a total of 52 cards in a deck of playing cards.
We are given that all the cards of 3 multiples have been removed.
We know that in a deck of 52 cards there will be 4 sets of 13 cards each in which two are black and two are red. Each set has 13 cards named numbers 2 to 10 and one ace, king, queen and jack.
Now, let us take one set of 13 cards
Here we can see that there are 3 cards of 3 multiples that are 3, 6 and 9 in each set.
We know that there are a total of 4 sets and in each set there are 3 cards of 3 multiples.
So, we can say that there are a total of 12 cards that are 3 multiples in the deck of 52 cards.
Let us assume that there are\['N'\] cards after removing the 3 multiples then we get
\[\begin{align}
  & \Rightarrow N=52-12 \\
 & \Rightarrow N=40 \\
\end{align}\]
We are given that a card at random is drawn from the remaining 40 cards.
We are asked to find the probability of getting an even number which is a red card.
We know that there will be two sets of red cards.
Now, let us take all the even numbers from the remaining cards for one set then we get the numbers as 2, 4, 8 and 10
Here, we can see that there are 2 sets of red cards and each set has 4 even numbers.
Let us assume that the possible outcomes of getting even number which is red as \['x'\] then we get
\[\begin{align}
  & \Rightarrow x=2\times 4 \\
 & \Rightarrow x=8 \\
\end{align}\]
Let us assume that the probability of getting even number which is red card as \[P\]
We know that the formula of probability is given as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
Now by using the above formula we get the required probability as
\[\Rightarrow P=\dfrac{x}{N}\]
By substituting the required values in the above formula we get
\[\begin{align}
  & \Rightarrow P=\dfrac{8}{40} \\
 & \Rightarrow P=\dfrac{1}{5} \\
\end{align}\]
Therefore the probability of getting an even number which is a red card after removing all the cards of 3 multiples is \[\dfrac{1}{5}\]

So, the correct answer is “Option c”.

Note: Students may make mistakes in taking the number of possible outcomes of getting an even number which is red.
We have the even numbers from the deck of 40 cards in each set as 2, 4, 8 and 10
Here, there are 4 even cards in each set.
But, students may take the cards as 2, 4, 6, 8 and 10
This is wrong because we already removed 6 from each set as a multiple of 3.
So, there are no 6 present in the remaining 40 cards. Therefore the number of possible outcomes will be
\[\begin{align}
  & \Rightarrow x=2\times 4 \\
 & \Rightarrow x=8 \\
\end{align}\]