A coin is biased so that the probability of falling head when tossed is `(1)/(4)` . If the coin is tossed 5 times , the probability of obtaining 2 heads and 3 tails is

Options

A. `(135)/(512)`
B. `(75)/(512)`
C. `(1)/(512)`
D. `(1)/(256)`

Answer

To find the probability of obtaining 2 heads and 3 tails when a biased coin is tossed 5 times, we can use the binomial probability formula. Here's the step-by-step solution:

### Step 1: Identify the probabilities
The probability of getting heads (success) when tossing the coin is given as:
\[ P(H) = \frac{1}{4} \]

The probability of getting tails (failure) is:
\[ P(T) = 1 - P(H) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 2: Determine the number of trials and successes
We are tossing the coin 5 times, so:
- Number of trials (n) = 5
- Number of successes (heads) we want (r) = 2

### Step 3: Use the binomial probability formula
The binomial probability formula is given by:
\[ P(X = r) = \binom{n}{r} P^r (1-P)^{n-r} \]
Where:
- \( \binom{n}{r} \) is the binomial coefficient, calculated as \( \frac{n!}{r!(n-r)!} \)
- \( P \) is the probability of success (getting heads)
- \( (1-P) \) is the probability of failure (getting tails)

### Step 4: Calculate the binomial coefficient
We need to calculate \( \binom{5}{2} \):
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the probabilities
Now we substitute the values into the formula:
\[ P(X = 2) = \binom{5}{2} \left(\frac{1}{4}\right)^2 \left(\frac{3}{4}\right)^{5-2} \]

Calculating \( P(X = 2) \):
\[ P(X = 2) = 10 \left(\frac{1}{4}\right)^2 \left(\frac{3}{4}\right)^3 \]

Calculating \( \left(\frac{1}{4}\right)^2 \):
\[ \left(\frac{1}{4}\right)^2 = \frac{1}{16} \]

Calculating \( \left(\frac{3}{4}\right)^3 \):
\[ \left(\frac{3}{4}\right)^3 = \frac{27}{64} \]

### Step 6: Combine the results
Now substituting back into the equation:
\[ P(X = 2) = 10 \times \frac{1}{16} \times \frac{27}{64} \]

Calculating:
\[ P(X = 2) = 10 \times \frac{27}{1024} = \frac{270}{1024} \]

### Step 7: Simplify the fraction
To simplify \( \frac{270}{1024} \):
Both numerator and denominator can be divided by 2:
\[ \frac{270 \div 2}{1024 \div 2} = \frac{135}{512} \]

### Final Answer
Thus, the probability of obtaining 2 heads and 3 tails when the biased coin is tossed 5 times is:
\[ \frac{135}{512} \]

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