# A standard deck of cards contains 52 cards. One card is selected from the deck.

• Compute the probability of randomly selecting a spade or diamond. P(spade or diamond)
• Compute the probability of randomly selecting a spade or diamond or heart. P(spade or diamond or heart)
• Compute the probability of randomly selecting a king or club. P(king or club)

This question aims to find the probability of different cards from a standard deck. Moreover, from the deck of 52 cards, one card is randomly selected.

Apart from that, the above question is based on the concept of statistics. Probability is simply how likely something is to happen, for example, a heads or tails result after a coin flip. In the same way, when a card is randomly selected, what are the chances or the probability that it is, for example, a spade or diamond.

The standard card decks have four different suits and 52 cards as a total. The four suits are heart, spades, diamonds, and clubs, and these suits have 13 cards each. The standard equation of probability is as follows:

$P ( A ) = \dfrac{\text{Number of favorable outcomes of A}}{\text{Total number of outcomes}}$

Therefore, the probability is calculated as follows:

$P(\text{spade or diamond)}$

$P(spade) = \dfrac{13}{52}$

$P(spade) = \dfrac{1}{4}$

$P(diamond) = \dfrac{13}{52}$

$P(diamond) = \dfrac{1}{4}$

So the probability of selecting a spade or a diamond is:

$\dfrac{1}{4} + \dfrac{1}{4} = \dfrac{1}{2} = 0.5$

$P(\text{Spade or Diamond or Heart})$

$P(heart) = \dfrac{13}{52}$

$P(heart) = \dfrac{1}{4}$

$P(spade) = \dfrac{13}{52}$

$P(spade) = \dfrac{1}{4}$

$P(diamond) = \dfrac{13}{52}$

$P(diamond) = \dfrac{1}{4}$

So the probability of selecting a spade, diamond or a heart is:

$\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4} = \dfrac{3}{4} = 0.75$

$P (\text{king or club) }$

$P(club) = \dfrac{13}{52}$

$P(club) = \dfrac{1}{4}$

Each suite contains a king; therefore, there are four kings in a deck of cards.
So the probability of selecting a king is:

$P(king) = \dfrac{4}{52}$

$P(king) = \dfrac{1}{13}$

Moreover, there is a card that is the king of the club; therefore, the probability of it is as follows:

$P(king of club) = \dfrac{1}{52}$

Hence, the probability of randomly selecting king or club is:

$P(king or club) = \dfrac{1}{4} + \dfrac{1}{13} – \dfrac{1}{52} = \dfrac{4}{13} = 0.308$

## Numerical Results

The probability of selecting a number is as follows.

$P(\text{spade or diamond)} = 0.5$

$P(\text{spade or diamond or heart)} = 0.75$

$P (\text{king or club) } = 0.308$

## Example

Find the probability of rolling a 4 when a dice is rolled.

Solution:

As a dice has six different numbers, therefore, by using the probability formula given above, $P(4)$ is calculated as:

$P(4) = \dfrac{4}{6}$

$= 0.667$

Images/ Mathematical drawings are created with Geogebra.
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