A card is drawn from a well shuffled pack of 52 cards. Find the probability that it is heart or a king .

Explanation

To find the probability that a card drawn from a well-shuffled pack of 52 cards is either a heart or a king, we can follow these steps:

### Step 1: Define the Events
Let:
- Event E = the card drawn is a heart.
- Event F = the card drawn is a king.

### Step 2: Calculate the Probability of Event E
There are 13 hearts in a deck of 52 cards. Therefore, the probability of drawing a heart (P(E)) is:
\[ P(E) = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4} \]

### Step 3: Calculate the Probability of Event F
There are 4 kings in a deck of 52 cards. Therefore, the probability of drawing a king (P(F)) is:
\[ P(F) = \frac{\text{Number of kings}}{\text{Total number of cards}} = \frac{4}{52} = \frac{1}{13} \]

### Step 4: Calculate the Probability of the Intersection of Events E and F
The intersection of events E and F (E ∩ F) is the event that the card drawn is the king of hearts. There is only 1 king of hearts in the deck. Therefore, the probability of drawing the king of hearts (P(E ∩ F)) is:
\[ P(E \cap F) = \frac{1}{52} \]

### Step 5: Apply the Addition Rule for Probabilities
To find the probability that the card drawn is either a heart or a king, we use the formula for the union of two events:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]
Substituting the values we calculated:
\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) = \frac{1}{4} + \frac{1}{13} - \frac{1}{52} \]

### Step 6: Find a Common Denominator
The least common multiple of 4, 13, and 52 is 52. We will convert each fraction to have a denominator of 52:
- \(\frac{1}{4} = \frac{13}{52}\)
- \(\frac{1}{13} = \frac{4}{52}\)
- \(\frac{1}{52} = \frac{1}{52}\)

### Step 7: Substitute and Simplify
Now substituting these values back into the equation:
\[ P(E \cup F) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{13 + 4 - 1}{52} = \frac{16}{52} \]

### Step 8: Simplify the Fraction
Now we simplify \(\frac{16}{52}\):
\[ P(E \cup F) = \frac{16 \div 4}{52 \div 4} = \frac{4}{13} \]

### Final Answer
Thus, the probability that a card drawn is either a heart or a king is:
\[ \frac{4}{13} \]

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