Probability Calculation Steps

Let $S_1$ be the event that the first student solves the problem, and $S_2$ be the event that the second student solves the problem.

We are given the probabilities:

• Probability of the first student solving: $P(S_1) = \frac{3}{5}$
• Probability of the second student solving: $P(S_2) = \frac{4}{5}$

We need to find the probability that at least one student solves the problem. This is represented as $P(S_1 \cup S_2)$.

It's often easier to calculate the probability of the complementary event: that *neither* student solves the problem. Let $\overline{S_1}$ be the event that the first student does not solve, and $\overline{S_2}$ be the event that the second student does not solve.

Calculating Complement Probabilities

The probabilities of the students *not* solving the problem are:

• $P(\overline{S_1}) = 1 - P(S_1) = 1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5}$
• $P(\overline{S_2}) = 1 - P(S_2) = 1 - \frac{4}{5} = \frac{5}{5} - \frac{4}{5} = \frac{1}{5}$

Probability of Neither Solving

Since the students solve the problem independently, the probability that neither solves it is the product of their individual probabilities of not solving:

$P(\overline{S_1} \cap \overline{S_2}) = P(\overline{S_1}) \times P(\overline{S_2})$

$P(\overline{S_1} \cap \overline{S_2}) = \frac{2}{5} \times \frac{1}{5} = \frac{2}{25}$

Final Probability Calculation

The probability that at least one student solves the problem is 1 minus the probability that neither solves it:

$P(\text{at least one solves}) = 1 - P(\text{neither solves})$

$P(S_1 \cup S_2) = 1 - P(\overline{S_1} \cap \overline{S_2})$

$P(S_1 \cup S_2) = 1 - \frac{2}{25} = \frac{25}{25} - \frac{2}{25} = \frac{23}{25}$