The probabilities that a student will receive an A, B, C or D grade in a certain course are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the student will receive at least a B grade (that is, either an A or a B)?

Introduction / Context:
This problem involves discrete probabilities for grade outcomes in a course. The student can receive a grade of A, B, C or D, each with specified probabilities. The question asks for the probability of receiving at least a B grade, which means the student gets either an A or a B. This introduces the idea of combining probabilities of multiple events using addition when the events are disjoint.

Given Data / Assumptions:

• Possible grades considered are A, B, C and D.
• Probability of A grade = 0.4.
• Probability of B grade = 0.3.
• Probability of C grade = 0.2.
• Probability of D grade = 0.1.
• These grades represent all possible outcomes for the course.

Concept / Approach:
The phrase at least a B grade means that the grade must be A or B. These two events are mutually exclusive, since the student cannot receive both grades at once. Therefore, we use the addition rule for disjoint events: the probability of A or B is the sum of the probabilities of A and B. No conditional probability or complex counting is required because the outcomes are already described with explicit probabilities.

Step-by-Step Solution:
Step 1: Identify the event of interest as receiving at least a B grade.Step 2: This event consists of two simple outcomes: getting an A or getting a B.Step 3: From the data, P(A) = 0.4 and P(B) = 0.3.Step 4: Because A and B are mutually exclusive, the probability of A or B is P(A) + P(B).Step 5: Add these to get 0.4 + 0.3 = 0.7.Step 6: Therefore, the probability that the student will receive at least a B grade is 0.7.
Verification / Alternative check:
We can check that the probabilities of all grade outcomes sum to 1, confirming that the distribution is valid. Adding 0.4, 0.3, 0.2 and 0.1 gives 1.0. The probability of not receiving at least a B grade is the probability of receiving either a C or a D, which is 0.2 + 0.1 = 0.3. The complement relationship P(at least B) + P(below B) = 0.7 + 0.3 = 1 confirms consistency with the full distribution.

Why Other Options Are Wrong:
0.21 is much smaller than the required 0.7 and does not correspond to adding any useful subset of given probabilities. 0.3 is the probability of receiving exactly a B grade, not at least a B grade. 0.5 corresponds to A or C, which is not the event described. 0.6 would require some other combination such as A plus C, and is not related to the phrase at least B grade.

Common Pitfalls:
Students sometimes misinterpret at least B as exactly B and use only 0.3. Another mistake is to include C in the sum, possibly due to confusion about grade ordering. It is also important to ensure that the events being added are disjoint; if overlapping events are added without correction, the total probability can exceed 1, which is impossible.

Final Answer:
The probability that the student receives at least a B grade is 0.7.