Question

Suppose that a student who is about to take a multiple choice test has only learned \(40 \%\) of the material covered by the exam. Thus, there is a \(40 \%\) chance that she will know the answer to a question. However, even if she does not know the answer to a question, she still has a \(20 \%\) chance of getting the right answer by guessing. If we choose a question at random from the exam, what is the probability that she will get it right?

Short answer

The probability that the student will get the question right is 52%.

Step-by-step solution

Calculating Probability of Knowing the Answer

First, determine the probability that the student knows the answer. This is given as 40%, or in probability notation, 0.40.

Calculating Probability of Correct Guess

Next, determine the probability that the student guesses the answer correctly. This is given as 20%, or in probability notation, 0.20.

Calculating Probability of Not Knowing the Answer

Calculate the probability that the student does not know the answer, which is the complement of knowing the answer. This is calculated as 1 - 0.40, which equals 0.60.

Combining Probabilities

Combine the probabilities using the law of total probability. The student can either know the answer (with probability of 0.40) or guess the answer correctly (with probability of 0.60 times 0.20). The total probability of getting the answer correct is thus: \(0.40 + (0.60 \times 0.20)\).

Calculating Total Probability

Carry out the calculations to get the total probability of getting the answer right: \(0.40 + (0.60 \times 0.20) = 0.40 + 0.12 = 0.52\). Therefore, the probability is 0.52 or 52%.

Key concepts

Law of Total Probability

Understanding the law of total probability is vital when dealing with complex problems that involve multiple events. It's like having a pie cut into different flavored slices, and you want to find out the probability of enjoying the pie regardless of the flavor. Imagine each event as a unique slice with its own likelihood.

The law of total probability helps us combine these chances to find the total probability of an outcome occurring. In our exercise, the student could correct a question by either knowing the answer or guessing it right. The law states that the total probability of a desired outcome is the sum of the probabilities of each distinct pathway leading to that outcome. In simpler terms, it's the probability of 'knowing the answer' plus the probability of 'not knowing but guessing right'.

When applying this rule, always ensure that the different pathways we consider cover all the possibilities and are mutually exclusive (don't overlap). For the student, this means we add up the chance of knowing the answer to the chance of correctly guessing when not knowing the answer. The law helps us to see the big picture probability by piecing together each part of the scenario.

Conditional Probability

Now, let's delve into conditional probability, which is like peeking into alternate timelines where certain conditions are already met. It tells us the probability of an event occurring given that another event has already happened. Think of it as the odds of finding a matching pair of socks assuming you've already picked a blue one.

In our problem, the condition is the student not knowing the answer to a question. Conditional probability allows us to calculate the likelihood of her guessing correctly under that specific condition. Mathematically, we denote this as P(Guessing right | Not knowing). To find this, we need to know the probability of guessing right, which is separate from the condition, and the probability of the condition itself.

It is crucial in multi-stage problems like ours, where an event (correctly guessing) depends on the previous event (not knowing the answer). By factoring in these dependencies, we ensure our probability calculation reflects the real-world scenario of the test-taking student.

Complementary Probability

Looking at complementary probability is akin to considering both sides of a coin; it tells us what the chances are of something not happening when we know the chances of it happening. For instance, if you have a 70% chance of rain, you consequently have a 30% chance of staying dry.

In our example, the complementary probability comes into play when we determine the probability that the student does not know the answer. This is simply one minus the probability of knowing the answer. It reflects the complete picture of possibilities for a particular situation — either something happens or it doesn't. Understanding this is crucial because it simplifies problems dealing with the 'chance of not happening', which is as essential as the 'chance of happening' in calculating overall probabilities.

Utilizing complementary probability is often the first step before diving into conditional probabilities, as it sets up the scenario under which other conditions can be explored. It's an invaluable tool for ensuring all possibilities are accounted for, providing us with the necessary other side of the probability equation.