A man draws a card from a pack one after another with replacement until he gets red card. The probability that he gets red card in `4^(th)` draw is

Options

A. `(1)/(16)`
B. `(1)/(8)`
C. `(1)/(4)`
D. `(1)/(2)`

Answer

To find the probability that a man draws a red card on the 4th draw while drawing cards with replacement, we can follow these steps:

### Step 1: Understand the Problem
We need to find the probability that the first three draws are black cards and the fourth draw is a red card.

### Step 2: Determine the Total Number of Cards
In a standard deck of cards, there are 52 cards in total, which includes:
- 26 red cards (hearts and diamonds)
- 26 black cards (clubs and spades)

### Step 3: Calculate the Probability of Drawing a Black Card
The probability of drawing a black card (not a red card) in one draw is:
\[ P(\text{Black}) = \frac{\text{Number of Black Cards}}{\text{Total Number of Cards}} = \frac{26}{52} = \frac{1}{2} \]

### Step 4: Calculate the Probability of Drawing a Red Card
The probability of drawing a red card in one draw is:
\[ P(\text{Red}) = \frac{\text{Number of Red Cards}}{\text{Total Number of Cards}} = \frac{26}{52} = \frac{1}{2} \]

### Step 5: Calculate the Probability of the Desired Sequence
To find the probability that the first three draws are black cards and the fourth draw is a red card, we multiply the probabilities of each event:
\[ P(\text{Black on 1st}) \times P(\text{Black on 2nd}) \times P(\text{Black on 3rd}) \times P(\text{Red on 4th}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \]

### Step 6: Simplify the Expression
Calculating the above expression gives:
\[ P = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

### Conclusion
The probability that he gets a red card on the 4th draw is:
\[ \frac{1}{16} \]

### Final Answer
The correct option is \(\frac{1}{16}\).

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