Probability question help?

The easiest way to answer this question is to draw a Venn diagram and visualise the various probabilities.

It would be easiest to draw one and then refer you to it but it's gonna take too much time. So basically draw a rectangle which represents the entire sample space. Now draw a shape inside the box which takes up 69% of the space. This represents the 0.69 probability of being a student taking mechanics. Now draw another shape, which intersects the first shape such that the intersection represents about 35% of the sample space, and which also in itself takes up 54% of the sample space. This represents the 0.54 probability of being a student doing maths. The intersection of these shapes should take up about 35% of the sample space, as we know there is a 0.35 probability that a student is taking both maths AND stats.

Now, there should be a small amount of space left over, because we know, from our answer to the first question, that the probability that a student takes maths or mechanics is NOT 1 (if it was, the 3 shapes would have to together take up the whole sample space).

Your answer to a is right - the probability that a student took maths OR mechanics, i.e. P(mathsUNIONmechanics) is equal to the total area of the first and second shapes within the Venn diagram. We don't actually know this shape directly, but we can calculate it. As you look at it, you can see that it must be P(maths) + P(mech) - P(mathsINTERSECTmech), i.e. MINUS the shape of intersection (if we don't do this, this shape is counted twice).

Thus: 0.69+0.54-0.35 = 0.88.

Note that the probability we've calculated is the probability that a student is doing maths OR mechanics. This gives us the solution to b, which is the probability that a student is NOT taking maths or mechanics. This is equivalent to the complement of the probability we just calculated.

There is an 0.88 probability that a student is taking maths or mechanics. This means that there is a 1-0.88 probability that a student is NOT taking maths or mechanics = 0.12.

Finally, use the Venn diagram again for part c. We want to find the union of the two events - 1. that a student takes mechanics, and 2. that a student doesn't take mathematics.

Look at the Venn diagram and shade in the area that represents a student NOT taking maths. You should shade the entire rectangle such that the shape representing taking maths is left white. This space represents NOT taking maths. Now, we want to find the intersection of this shape with the shape which represents taking mechanics. Thus we can forget about the whole shaded area outside the mechanics shape. We are interested in the intersection of the two shapes. We can see that we have to count the whole of the mechanics shape, MINUS the bit of the mechanics shape which is also part of the maths shape.

We know the probability associated with this shape - its 0.35. Thus the answer to c) is 0.69 - 0.35 = 0.34.

Well considering the student base has a 58% discrepancy i think the answer to all of these is questionable and suddenly takes an existentialist turn as the student might not exist. So the answer to c. is at the same moment one of two answer 69% chance or 43.67% chance with an 18.97% possibility of being imaginary.

b =12

c= 34

a) 0.88

b) 0.12

c) 19/100