In a lottery, 16 tickets are sold and 4 prizes are awarded. If a person buys 4 tickets,what is the probability of his winning a prize?

Options

A. `4/16^(4)`
B. `175/256`
C. `1/4`
D. `81/256`

Answer

To find the probability of a person winning a prize in a lottery where 16 tickets are sold and 4 prizes are awarded, and the person buys 4 tickets, we can follow these steps:

### Step 1: Identify the total number of tickets and prizes
- Total tickets sold = 16
- Total prizes = 4

### Step 2: Determine the number of tickets bought by the person
- Tickets bought by the person = 4

### Step 3: Calculate the probability of winning at least one prize
To find the probability of winning at least one prize, we can use the complementary probability approach. First, we calculate the probability of not winning any prize and then subtract that from 1.

1. **Calculate the total ways to choose 4 tickets from 16:**
\[ \text{Total ways to choose 4 tickets from 16} = \binom{16}{4} \]

2. **Calculate the ways to choose 4 tickets from the 12 tickets that do not win a prize:**
\[ \text{Ways to choose 4 tickets from 12} = \binom{12}{4} \]

3. **Calculate the probability of not winning any prize:**
\[ P(\text{not winning}) = \frac{\binom{12}{4}}{\binom{16}{4}} \]

4. **Calculate the probability of winning at least one prize:**
\[ P(\text{winning at least one}) = 1 - P(\text{not winning}) \]

### Step 4: Perform the calculations
1. Calculate \(\binom{16}{4}\):
\[ \binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820 \]

2. Calculate \(\binom{12}{4}\):
\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

3. Calculate \(P(\text{not winning})\):
\[ P(\text{not winning}) = \frac{495}{1820} \]

4. Calculate \(P(\text{winning at least one})\):
\[ P(\text{winning at least one}) = 1 - \frac{495}{1820} = \frac{1820 - 495}{1820} = \frac{1325}{1820} \]

5. Simplify \(\frac{1325}{1820}\):
\[ \frac{1325 \div 65}{1820 \div 65} = \frac{25}{28} \]

### Final Answer
The probability of the person winning at least one prize is \(\frac{25}{28}\).