Question

Four persons independently solve a certain problem correctly with probabilities \(\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}\). Then the probability that the problem is solved correctly by at least one of them is (A) \(\frac{235}{256}\) (B) \(\frac{21}{256}\) (C) \(\frac{3}{256}\) (D) \(\frac{253}{256}\)

Short answer

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

Step-by-step solution

- Calculate the probability of each person failing

To find the probability of at least one person solving the problem correctly, it's easier to first calculate the probability that each person fails to solve the problem. These are the complements of the given probabilities. For the first person, the probability of failing is 1 - 1/2 = 1/2. For the second person, it's 1 - 3/4 = 1/4. For the third person, it's 1 - 1/4 = 3/4. And for the fourth person, it's 1 - 1/8 = 7/8.

- Calculate the combined probability of everyone failing

Now that we have the probabilities of each person failing, we assume independence to multiply these probabilities together to find the probability that all four fail. Multiplying 1/2, 1/4, 3/4, and 7/8 gives (1/2) * (1/4) * (3/4) * (7/8) = 21/256. This is the probability that none of the four persons solve the problem correctly.

- Calculate the probability of at least one solving correctly

To find the probability that at least one person solves the problem, subtract the probability that no one solves it from 1. This gives us the complement of everyone failing. Thus, the probability of at least one solving it correctly is 1 - 21/256 = (256/256) - (21/256) = 235/256.

Key concepts

Independent Events in Probability

The concept of independent events is a cornerstone in understanding probability. Independent events are those whose occurrence does not affect the probability of occurrence of another event. For example, if you flip a coin, the result of the flip has no impact on the outcome of the next flip; each flip is independent of the other.

In the context of our exercise, the probability of each person solving the problem correctly does not depend on whether the others solve it or not. This is taken into account when we calculate the combined probability of all four individuals failing to solve the problem. When events are independent, the probability of all events occurring is the product of their individual probabilities. As such, in our calculations, we multiply the probability of each person failing to find the overall probability of all failing.

Probability Complements

Understanding probability complements is crucial for simplifying complex probability problems. The complement of an event is the event not happening. In terms of probability, it is calculated as 1 minus the probability of the event occurring. Put simply, if the probability of an event A happening is P(A), then the probability of A not happening, denoted as P(A'), is 1 - P(A).

In our exercise, we're interested in the probability of at least one person solving the problem correctly. To make it easier, we first consider the probability of each person not solving the problem, which are the complements of the given probabilities. After finding these complements, we use them to calculate the combined probability that no one solves the problem. Finally, we find the desired probability through the complement of this combined probability.

Combining Probabilities

When faced with multiple components in a probability scenario, knowing how to combine probabilities is essential. There are different methods of combining probabilities depending on the events' relationship to one another. If events are independent, as previously mentioned, we multiply their probabilities together to find the probability of all events occurring consecutively.

But what about the probability of at least one event occurring? This is where the complement rule is handy, especially when the events are independent. By finding the probability that no event occurs and subtracting it from 1, we efficiently determine the probability of at least one event happening, without having to add the probabilities of all possible combinations of events occurring individually. In our solved exercise, this method streamlines finding the probability that at least one person solves the problem correctly.