Question

One card is drawn from a well-shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the drawn card is
(i) A diamond
(ii) Not an ace.

Answer 1

Hint: For solving this problem we need to find the possible chances of the outcome given in each question and then we use the formulae of probability as
\[P=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]. We know that in a deck of 52 cards there are 4 parts of 13 cards named diamond, hearts, spades, and clubs. Each part has numbers 2-10, one ace, king, queen, and Jack. By using this information we calculate the required probability.

Complete step-by-step solution
Let us solve the first question, which is getting a diamond card.
(i) Let us assume that \[E\] be the event of getting a diamond card from a pack of 52 cards.
We know that in a deck of 52 cards there are 4 parts of 13 cards named diamond, hearts, spades, and clubs.
So, the number of possible outcomes of getting a diamond card is ‘13’
Here, we are given to draw one card. So, the total number of outcomes will be ‘52’.
We know that the probability formula is given as
\[P\left( E \right)=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
By substituting the required values in above equation we will get
\[\begin{align}
  & \Rightarrow P\left( E \right)=\dfrac{13}{52} \\
 & \Rightarrow P\left( E \right)=\dfrac{1}{4} \\
\end{align}\]
Therefore the probability of getting a diamond when one card is drawn from a deck of 52 cards is \[\dfrac{1}{4}\].
Now let us go to the second question: getting a card which is not an ace.
(ii) Let us assume that \[E\] be the event of getting a card that is not an ace from a pack of 52 cards.
We know that in a deck of 52 cards there are 4 parts of 13 cards named diamond, hearts, spades, and clubs. Each part has numbers 2-10, one ace, king, queen, and Jack.
So, the number of possible outcomes of getting a card that is not an ace is \[52-4=48\]
Here, we are given to draw one card. So, the total number of outcomes will be ‘52’.
We know that the probability formula is given as
\[P\left( E \right)=\dfrac{\text{number of possible outcomes}}{\text{total number of outcomes}}\]
By substituting the required values in above equation we will get
\[\begin{align}
  & \Rightarrow P\left( E \right)=\dfrac{48}{52} \\
 & \Rightarrow P\left( E \right)=\dfrac{12}{13} \\
\end{align}\]
Therefore the probability of getting a card that is not an ace when one card is drawn from a deck of 52 cards is \[\dfrac{12}{13}\].

Note: Students may make mistakes in solving the second part. There the total number of possible outcomes is ‘48’ because it was mentioned that the card is not an ace. But in a hurry students will think that they asked for a card drawn is an ace and take possible outcomes as ‘4’ which in turn will get the wrong answer. Reading the question carefully is important.