The probability that a first year student entering a certain private college needs neither a developmental math course nor a developmental English is 64% while 25% require a developmental math course and 30% require a developmental English course. Find the probability that a first year student requires both a development math course and a developmental English course.
Question
The probability that a first year student entering a certain private college needs neither a developmental math course nor a developmental English is 64% while 25% require a developmental math course and 30% require a developmental English course. Find the probability that a first year student requires both a development math course and a developmental English course.
Solution
To solve this problem, we can use the formula for the probability of the union of two events, which is given by:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Where:
- P(A ∪ B) is the probability that either event A occurs, or event B occurs, or they both occur.
- P(A) is the probability that event A occurs.
- P(B) is the probability that event B occurs.
- P(A ∩ B) is the probability that both events A and B occur.
In this case, we know that:
- The probability that a student needs neither a developmental math course nor a developmental English course is 64%, so the probability that a student needs either a developmental math course or a developmental English course (or both) is 100% - 64% = 36%.
- The probability that a student needs a developmental math course (event A) is 25%.
- The probability that a student needs a developmental English course (event B) is 30%.
We want to find P(A ∩ B), the probability that a student needs both a developmental math course and a developmental English course. We can rearrange the formula to solve for this:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
Substituting the known values:
P(A ∩ B) = 25% + 30% - 36% = 19%
So, the probability that a first year student requires both a developmental math course and a developmental English course is 19%.
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