In an entrance test that is graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both?
Let A be the event that the student passes the first examination and B be the event that the student passes the second examination. We are given the probabilities for passing each subject and the probability of passing at least one.
Step 1: Identify the given probabilities from the question. Probability of passing the first examination, P(A) = 0.8. Probability of passing the second examination, P(B) = 0.7. Probability of passing at least one of them, P(A or B) = 0.95.
Step 2: Apply the formula for the probability of 'A or B'. We know that P(A or B) = P(A) + P(B) – P(A and B). Substituting the given values into the formula: 0.95 = 0.8 + 0.7 – P(A and B) 0.95 = 1.5 – P(A and B)
Step 3: Solve for the probability of passing both. Rearranging the equation to find P(A and B): P(A and B) = 1.5 – 0.95 P(A and B) = 0.55 Therefore, the probability of passing both examinations is 0.55.
Explanation
This solution corresponds to Question 19 found in the provided context. It applies the Addition Theorem of Probability, which is the relevant concept for finding the intersection of two events (passing both) when the individual probabilities and the union probability (passing at least one) are known. The calculation follows the standard formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Solution Steps
Step 1: Define events A and B, and list the given values: P(A) = 0.8, P(B) = 0.7, and P(A or B) = 0.95.
Step 2: Use the addition rule of probability: P(A or B) = P(A) + P(B) - P(A and B).
Step 3: Substitute the values: 0.95 = 0.8 + 0.7 - P(A and B).
Step 4: Solve the equation: P(A and B) = 1.5 - 0.95 = 0.55.