Black-Scholes-Merton formulas for the prices of European call and put options:
$C = S_O N(d_1) - K e^{(rT)} N(d_2)$

The term $N(d_2)$ in the equation is the probability that a call option will be exercised in a risk-neutral world. (Note: The $N(d_1)$ term is not quite so easy to interpret. The expression $S_O N(d_1) e^{rT}$ is the expected stock price at time T in a risk-neutral world when stock prices less than the strike price are counted as zero.) The strike price is only paid if the stock price is greater than K, and this has a probability of $N(d_2)$. Therefore:

Probability of exercising the option = $N(d_2)$

$$d_1 = \frac{\ln\left(\frac{S_0}{K}\right) + \left(r + \frac{\sigma^2}{2}\right) \times T}{\sigma \times \sqrt{T}} = \frac{\ln\left(\frac{50}{48}\right) + \left(8\% + \frac{25\%^2}{2}\right) \times 0.5}{25\% \times \sqrt{0.5}} = 0.5456$$

Now, we calculate $d_2$: $d_2 = d_1 - \sigma \times \sqrt{T} = 0.5456 - 0.25 \times \sqrt{0.5} = 0.5456 - 0.1768 = 0.3688$

The probability of exercise is $N(d_2)$, which can be found using the standard normal cumulative probability table: $N(0.3688) \approx 0.6441$