. The probability that a student passes in Statistics test is 2/3 and the probability that he passes both Statistics and Mathematics test is 14/45.The probability that he passes at least one test is 4/5.What is the probability that the student passes the Mathematics test?
Let's solve this step-by-step:
1. Let's assume that the probability of passing the Statistics test is P(S) = 2/3.
2. The probability of passing both the Statistics and Mathematics tests is given as P(S ∩ M) = 14/45.
3. The probability of passing at least one test is given as P(S ∪ M) = 4/5.
Now, we will use these probabilities to find the probability of passing the Mathematics test.
4. To find the probability of passing at least one test, we can use the formula: P(S ∪ M) = P(S) + P(M) - P(S ∩ M).
So, substituting the given values, we have: 4/5 = 2/3 + P(M) - 14/45.
5. Simplifying this equation, we get: P(M) = 4/5 + 14/45 - 2/3.
6. Finding a common denominator, we get: P(M) = (36/45) + (14/45) - (30/45).
7. Combining the fractions, we have: P(M) = 20/45.
8. Simplifying further, we get: P(M) = 4/9.
Therefore, the probability that the student passes the Mathematics test is 4/9.
To find the probability that the student passes the Mathematics test, we can use the concept of conditional probability.
Let's denote the event of passing the Statistics test as A and the event of passing the Mathematics test as B. We are given the following:
- P(A) = 2/3 (probability of passing the Statistics test)
- P(A∩B) = 14/45 (probability of passing both Statistics and Mathematics test)
- P(A∪B) = 4/5 (probability of passing at least one test)
We can use these pieces of information to find the probability of passing the Mathematics test.
First, let's find the probability of not passing both tests (Statistics and Mathematics): P(A'∩B').
Since P(A∪B) = P(A) + P(B) - P(A∩B), we can rearrange the equation to find P(A∩B):
P(A∩B) = P(A) + P(B) - P(A∪B)
14/45 = 2/3 + P(B) - 4/5
Simplifying:
14/45 = 10/15 + P(B) - 4/5
14/45 = 2/15 + P(B)
Now, let's find the probability of not passing the Mathematics test: P(B').
Since P(B') = 1 - P(B), we can substitute 1 - P(B) for P(B'):
P(B') = 1 - P(B)
P(B') = 1 - (2/15 + P(B))
P(B') = 1 - 2/15 - P(B)
Using the property that P(A'∩B') = 1 - P(A∪B), we can substitute P(B') for P(A'∩B'):
P(B') = 1 - P(A∪B)
P(B') = 1 - 4/5
Simplifying:
P(B') = 1/5
Finally, let's find the probability of passing the Mathematics test, P(B):
P(B) = 1 - P(B')
P(B) = 1 - 1/5
P(B) = 4/5
Therefore, the probability that the student passes the Mathematics test is 4/5.