A radioactive sample has half-life of `5` years. Probability of decay in `10` years will be.

Options

A. 1
B. 0.75
C. 0.5
D. 0.25

Explanation

To solve the problem of finding the probability of decay of a radioactive sample over a period of 10 years, given its half-life of 5 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**:
The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life (T) is given as 5 years.

2. **Determine the Time Period**:
We need to find the probability of decay over a time period (t) of 10 years.

3. **Use the Decay Formula**:
The fraction of undecayed particles after time t can be calculated using the formula:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{T}} \]
where \(N_0\) is the initial quantity of the substance, \(N\) is the quantity remaining after time t, and T is the half-life.

4. **Substitute the Values**:
Here, \(t = 10\) years and \(T = 5\) years. Plugging these values into the formula gives:
\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^{\frac{10}{5}} = \left(\frac{1}{2}\right)^{2} = \frac{1}{4} \]

5. **Calculate the Decayed Fraction**:
The fraction of the sample that has decayed is given by:
\[ \text{Decayed fraction} = 1 - \frac{N}{N_0} = 1 - \frac{1}{4} = \frac{3}{4} \]

6. **Convert to Percentage**:
To express the decayed fraction as a percentage, multiply by 100:
\[ \text{Probability of decay} = \frac{3}{4} \times 100 = 75\% \]

### Final Answer:
The probability of decay in 10 years is **75%**.

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