From the pack of 52 playing cards, the black face cards are removed. Now the cards are re-shuffled and then a card is drawn from the remaining pack of cards. Find the probability that the card drawn is :
(iv) a spade card

Answer

To find the probability of drawing a spade card after removing the black face cards from a standard deck of 52 playing cards, we can follow these steps:

### Step 1: Identify the total number of cards in the deck
A standard deck of playing cards contains 52 cards.

**Hint:** Remember that a standard deck consists of four suits: hearts, diamonds, clubs, and spades, each containing 13 cards.

### Step 2: Determine the black face cards
The black face cards in a deck are the King, Queen, and Jack of spades and clubs. Therefore, there are a total of 6 black face cards (3 from spades and 3 from clubs).

**Hint:** Consider how many face cards are in each suit and which suits are black.

### Step 3: Calculate the new total number of cards after removing the black face cards
After removing the 6 black face cards from the original 52 cards, the remaining number of cards is:
\[ 52 - 6 = 46 \]

**Hint:** Subtract the number of removed cards from the total number of cards in the deck.

### Step 4: Determine the number of spade cards remaining
Initially, there are 13 spade cards in the deck. Since we removed the 3 black face cards from the spades, the number of spade cards remaining is:
\[ 13 - 3 = 10 \]

**Hint:** Think about how many spade cards are face cards and how many remain after the removal.

### Step 5: Calculate the probability of drawing a spade card
The probability \( P \) of drawing a spade card is given by the ratio of the number of favorable outcomes (spade cards remaining) to the total number of possible outcomes (total cards remaining):
\[ P(\text{spade}) = \frac{\text{Number of spade cards remaining}}{\text{Total number of cards remaining}} = \frac{10}{46} \]

### Step 6: Simplify the probability
We can simplify the fraction:
\[ P(\text{spade}) = \frac{10 \div 2}{46 \div 2} = \frac{5}{23} \]

**Hint:** Always check if the fraction can be simplified by finding the greatest common divisor.

### Final Answer
The probability that the card drawn is a spade card is:
\[ \frac{5}{23} \]