Question

A card is drawn at random from a well-shuffled deck of $52$ playing cards. Find the probability that the card drawn is a card of spade or an ace, a red king, neither a king nor a queen, and either a king or queen.

Answer 1

Step 1: Write the formula for probability and find the total number of outcomes:

The formula for probability is,

$\text{Probability}=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

Given that, a card is drawn from a well-shuffled deck of $52$ playing cards.

Thus, the total number of outcomes will be $52.$

Step 2: Find the probability that the card drawn is a card of spade or an ace:

There are $13$spades including its ace, and $3$ other aces are their other suits

Thus, the probability of getting a spade or ace is

$$\begin{gathered} =\frac{(13+3)}{52} \\ =\frac{16}{52} \\ =\frac{4}{13} \end{gathered}$$

​Step 3: Find the probability that the card drawn is a red king:

There are two red kings.

So the probability

$$\begin{gathered} =\frac{2}{52} \\ =\frac{1}{26} \end{gathered}$$

​Step 4: Find the probability that the card drawn is neither a king nor a queen:

There are $4$ queens and $4$ kings.

Thus, the cards which are neither queen nor king $=52-8=44$

Probability is

$$\begin{gathered} =\frac{44}{52} \\ =\frac{11}{13} \end{gathered}$$

​Step 5: Find the probability that the card drawn is either a king or queen:

There are $4$ kings and $4$ queens.

So the probability is

$$\begin{gathered} =\frac{8}{52} \\ =\frac{2}{13} \end{gathered}$$
Hence, from the above conclusions, the probability that the card drawn is a card of spade or an ace is $\frac{4}{13}$, a red king is $\frac{1}{26}$, neither a king nor a queen is $\frac{11}{13}$, and either a king or queen is $\frac{2}{13}$