A biased coin in which probability of getting head is twice to that of tail. If coin is tossed 3 times then the probability of getting two tails and one head is

Options

A. `1/9`
B. `2/9`
C. `2/(27)`
D. `1/(27)`

Answer

To solve the problem step by step, we need to find the probability of getting two tails and one head when a biased coin is tossed three times.

### Step 1: Define the probabilities
Let the probability of getting a tail be \( q \) and the probability of getting a head be \( p \). According to the problem, the probability of getting a head is twice that of getting a tail. Therefore, we can write:

\[ p = 2q \]

### Step 2: Use the total probability
Since the total probability must equal 1, we can write:

\[ p + q = 1 \]

### Step 3: Substitute \( p \) into the total probability equation
Now, substitute \( p \) from Step 1 into the total probability equation:

\[ 2q + q = 1 \]

This simplifies to:

\[ 3q = 1 \]

### Step 4: Solve for \( q \)
Now, solve for \( q \):

\[ q = \frac{1}{3} \]

### Step 5: Find \( p \)
Now that we have \( q \), we can find \( p \):

\[ p = 2q = 2 \times \frac{1}{3} = \frac{2}{3} \]

### Step 6: Calculate the probability of getting 2 tails and 1 head
To find the probability of getting exactly 2 tails and 1 head in 3 tosses, we can use the binomial probability formula:

\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]

In our case, we want \( k = 1 \) head (which means \( n - k = 2 \) tails), and \( n = 3 \):

\[ P(2 \text{ tails and } 1 \text{ head}) = \binom{3}{2} p^1 q^2 \]

### Step 7: Calculate the binomial coefficient
The binomial coefficient \( \binom{3}{2} \) is:

\[ \binom{3}{2} = 3 \]

### Step 8: Substitute the values of \( p \) and \( q \)
Now substitute \( p \) and \( q \):

\[ P(2 \text{ tails and } 1 \text{ head}) = 3 \cdot \left(\frac{2}{3}\right)^1 \cdot \left(\frac{1}{3}\right)^2 \]

### Step 9: Calculate the probability
Now calculate the probability:

\[ = 3 \cdot \frac{2}{3} \cdot \frac{1}{9} = 3 \cdot \frac{2}{27} = \frac{6}{27} = \frac{2}{9} \]

### Final Answer
Thus, the probability of getting 2 tails and 1 head when tossing the coin 3 times is:

\[ \frac{2}{9} \]

---