Q14(v):
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (v) a spade

Solution :

Given: A well-shuffled deck of $52$ playing cards.

To find: The probability of drawing a card that is a spade.

Visual Representation of Card Suits:

Step 1: Define the Sample Space

The total number of possible outcomes when drawing one card from a well-shuffled deck is the total number of cards in the deck.

Let $n(S)$ be the total number of outcomes.

$n(S) = 52$ [Since a standard deck contains 52 cards]

Step 2: Identify the Favorable Outcomes

A standard deck of cards is divided into four suits: Spades, Hearts, Diamonds, and Clubs. Each suit contains exactly $13$ cards.

Let $E$ be the event of drawing a spade.

The number of favorable outcomes $n(E)$ is the number of spade cards in the deck.

$n(E) = 13$ [Since there are 13 cards of the spade suit]

Step 3: Apply the Probability Formula

The probability of an event $P(E)$ is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

$P(E) = \frac{n(E)}{n(S)}$ [Definition of theoretical probability]

Step 4: Perform the Calculation

Substitute the values identified in Step 1 and Step 2 into the formula:

$P(\text{Spade}) = \frac{13}{52}$

To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is $13$:

$P(\text{Spade}) = \frac{13 \div 13}{52 \div 13}$

$P(\text{Spade}) = \frac{1}{4}$

Final Answer: The probability of drawing a spade is $\frac{1}{4}$.

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