1. Use the following information about a group of college students: Fifty percent are taking Algebra. Sixty percent are taking Biology. Thirty percent are taking both Biology and Algebra. A student from the group is selected at random, calculate the following:
a) The probability that is taking Algebra or Biology.
b) The probability that is taking Biology given that is taking Algebra.
To answer these questions, we can use the concept of probabilities and set notation. Let's break it down step by step.
Given information:
- 50% are taking Algebra.
- 60% are taking Biology.
- 30% are taking both Biology and Algebra.
a) The probability that a student is taking Algebra or Biology can be calculated using the principle of inclusion-exclusion. We sum the probabilities of taking Algebra and Biology separately and then subtract the probability of taking both:
P(Algebra or Biology) = P(Algebra) + P(Biology) - P(Algebra and Biology)
To find the probability of taking Algebra or Biology:
P(Algebra) = 50% = 0.5
P(Biology) = 60% = 0.6
P(Algebra and Biology) = 30% = 0.3
P(Algebra or Biology) = 0.5 + 0.6 - 0.3 = 0.8
Therefore, the probability that a student is taking Algebra or Biology is 0.8 or 80%.
b) The probability that a student is taking Biology given that they are taking Algebra can be calculated using conditional probability. The formula for this conditional probability is:
P(Biology|Algebra) = P(Biology and Algebra) / P(Algebra)
We already know P(Algebra and Biology) = 0.3 and P(Algebra) = 0.5.
P(Biology|Algebra) = 0.3 / 0.5 = 0.6
Therefore, the probability that a student is taking Biology given that they are taking Algebra is 0.6 or 60%.