# To find the probability that a card drawn at random from a pack of 52 cards is neither a heart nor a king?

(a)$\dfrac{4}{13}$

(b)$\dfrac{9}{13}$

(c) $\dfrac{2}{13}$

(d) $\dfrac{5}{13}$

Last updated date: 26th Mar 2023

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Answer

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Hint: We start by finding the total possible number of outcomes possible for this problem. Since 1 card is drawn at random from a pack of 52 cards, the total possible number of outcomes is 52.

Now, to calculate the total number of desirable outcomes, we need to count the total number of cards from a pack of 52 cards that are neither a heart nor a king.

Complete step-by-step answer:

To calculate this, we know that there are four suits of cards – clubs, diamonds, spades and hearts. Each suit contains 13 cards. (totalling up to 52 cards)

Now, according to the question, since the card is not a heart, it belongs to the remaining 3 suits (clubs, diamonds and spades). Out of these, since the card is not a king either, there are 12 desirable outcomes in each of these 3 suits (since one of the cards in the suits is a king, we remove this outcome). Now, we get the desired number of outcomes as $12\times 3=36$.

Now,

$\begin{align}

& \text{probability = }\dfrac{\text{total desirable outcomes}}{\text{total possible outcomes}} \\

& \text{probability = }\dfrac{\text{36}}{\text{52}} \\

& \text{probability = }\dfrac{\text{9}}{13} \\

\end{align}$

Hence, the correct answer is (b) $\dfrac{9}{13}$.

Note: An alternative way to solve the problem is to subtract the possible number of outcomes for which a card drawn is a heart and king from 1. To explain,

Probability (card is heart and king) + Probability (card is neither a heart nor king) = 1

Thus,

Probability (card is neither a heart nor king) = 1 - Probability (card is heart and king)

Now, for the card to belong to hearts suit, there are 13 possible outcomes. Further, for a card to be a king, there are 4 possible outcomes. However, out of these 4 outcomes, 1 of the outcomes is common with 13 outcomes of heart suit. (thus, this outcome is removed). We are thus left with 13+4-1=16 outcomes.

Thus,

$\begin{align}

& \text{probability = }\dfrac{\text{16}}{\text{52}} \\

& \text{probability = }\dfrac{4}{13} \\

\end{align}$

Thus, Probability (card is neither a heart nor king) = $1-\dfrac{4}{13}=\dfrac{9}{13}$

Now, to calculate the total number of desirable outcomes, we need to count the total number of cards from a pack of 52 cards that are neither a heart nor a king.

Complete step-by-step answer:

To calculate this, we know that there are four suits of cards – clubs, diamonds, spades and hearts. Each suit contains 13 cards. (totalling up to 52 cards)

Now, according to the question, since the card is not a heart, it belongs to the remaining 3 suits (clubs, diamonds and spades). Out of these, since the card is not a king either, there are 12 desirable outcomes in each of these 3 suits (since one of the cards in the suits is a king, we remove this outcome). Now, we get the desired number of outcomes as $12\times 3=36$.

Now,

$\begin{align}

& \text{probability = }\dfrac{\text{total desirable outcomes}}{\text{total possible outcomes}} \\

& \text{probability = }\dfrac{\text{36}}{\text{52}} \\

& \text{probability = }\dfrac{\text{9}}{13} \\

\end{align}$

Hence, the correct answer is (b) $\dfrac{9}{13}$.

Note: An alternative way to solve the problem is to subtract the possible number of outcomes for which a card drawn is a heart and king from 1. To explain,

Probability (card is heart and king) + Probability (card is neither a heart nor king) = 1

Thus,

Probability (card is neither a heart nor king) = 1 - Probability (card is heart and king)

Now, for the card to belong to hearts suit, there are 13 possible outcomes. Further, for a card to be a king, there are 4 possible outcomes. However, out of these 4 outcomes, 1 of the outcomes is common with 13 outcomes of heart suit. (thus, this outcome is removed). We are thus left with 13+4-1=16 outcomes.

Thus,

$\begin{align}

& \text{probability = }\dfrac{\text{16}}{\text{52}} \\

& \text{probability = }\dfrac{4}{13} \\

\end{align}$

Thus, Probability (card is neither a heart nor king) = $1-\dfrac{4}{13}=\dfrac{9}{13}$

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