Final answer:
The probability that the student passes the English test is calculated using the given probabilities for passing the physics test, passing both tests, and passing at least one test. After manipulating these probabilities, the probability of passing the English test is found to be 4/9.
Step-by-step explanation:
To find the probability that the student passes the English test, we can use the given probabilities for passing each exam and passing both exams. Let P(Physics) be the probability of passing the physics test, P(English) be the probability of passing the English test, and P(Both) be the probability of passing both tests.
We are given the following:
- P(Physics) = 2/3
- P(Both) = 14/45
- The probability of passing at least one test (Physics or English or both) is 4/5
Using these probabilities, we can calculate the probability of passing the English test by using the formula for the probability of the union of two events:
P(Physics or English) = P(Physics) + P(English) - P(Both)
Substituting the known values:
4/5 = (2/3) + P(English) - (14/45)
To find P(English), we need to isolate it:
P(English) = 4/5 - 2/3 + 14/45
Before adding or subtracting fractions, we need a common denominator. The common denominator for 5, 3, and 45 is 45, so we convert each fraction:
= (36/45) - (30/45) + (14/45)
= (36 - 30 + 14) / 45
= 20/45
= 4/9
Therefore, the probability of passing the English test is 4/9.