Concept Explanation: Probability of At Least One Success

When dealing with independent events, the probability that at least one of them occurs can be found using the complement rule. Instead of calculating the probability for each possible case (1, 2, 3, or 4 persons solving it), it's easier to calculate the probability that none solve it correctly and subtract that from 1.

Step-by-Step Solution:

Step 1: Identify the probabilities of each person solving correctly:

Person 1: $P1=\frac{1}{2}$

Person 2: $P2=\frac{3}{4}$

Person 3: $P3=\frac{1}{4}$

Person 4: $P4=\frac{1}{8}$

Step 2: Find the probability that each person fails to solve the problem (the complement of their success probability).

Person 1 fails: $Q1=1-\frac{1}{2}=\frac{1}{2}$

Person 2 fails: $Q2=1-\frac{3}{4}=\frac{1}{4}$

Person 3 fails: $Q3=1-\frac{1}{4}=\frac{3}{4}$

Person 4 fails: $Q4=1-\frac{1}{8}=\frac{7}{8}$

Step 3: Since the persons are independent, the probability that all fail is the product of their individual failure probabilities.

$P\left(\text{None solve it}\right)=Q1\times Q2\times Q3\times Q4=\frac{1}{2}\times \frac{1}{4}\times \frac{3}{4}\times \frac{7}{8}$

Step 4: Calculate the product:

$\frac{1}{2}\times \frac{1}{4}=\frac{1}{8}$

$\frac{1}{8}\times \frac{3}{4}=\frac{3}{32}$

$\frac{3}{32}\times \frac{7}{8}=\frac{21}{256}$

So, $P\left(\text{None}\right)=\frac{21}{256}$

Step 5: Apply the complement rule to find the probability that at least one person solves it correctly.

$P\left(\text{At least one}\right)=1-P\left(\text{None}\right)=1-\frac{21}{256}=\frac{256}{256}-\frac{21}{256}=\frac{235}{256}$

Final Answer: The probability that the problem is solved correctly by at least one of them is $\frac{235}{256}$.

Related Topics:

• Probability Theory
• Independent Events
• Complement Rule
• Multiplication Rule for Independent Events

Key Formulae and Theory:

Complement Rule: For any event A, the probability that A occurs is 1 minus the probability that A does not occur. $P\left(A\right)=1-P\left(\text{not}A\right)$

Multiplication Rule for Independent Events: If events A, B, C,... are independent, the probability that they all occur is the product of their individual probabilities. $P\left(A\text{and}B\text{and}C\text{and ...}\right)=P\left(A\right)\times P\left(B\right)\times P\left(C\right)\times \dots$

Note: This approach is much more efficient than calculating the probability for every possible combination of successes (1 success, 2 successes, etc.), especially as the number of events increases.