We are given that a sample of 1046 people were asked if they supported, opposed, or were unsure about an education tax. The results are as follows:
(a) What is the probability that a randomly chosen person is opposed or female?
Since the two events are not mutually exclusive (e.g., it is possible to be opposed to the tax and to be a woman) we use the additive rule:
P(A or B)=P(A)+P(B)-P(A and B). This formula takes into account the overlap between the two sets.
So, P(O or F)=P(O)+P(F)-P(O and F)=`620/1046+550/1046-292/1046=878/1046~~.839`.
An alternative method is to use the definition of probability. The size of the event space is the number of people who have the given attributes opposed to the tax or female; |E|=328+292+237+21=878, while the size of the sample space is 1046. Then, `P=(|E|)/(|S|)=878/1046~~.839` (note that to get the event space, we added the numbers in the opposed column to the numbers in the female row, only adding 292 (female and opposed) once).
(b) What is the probability that a randomly chosen individual supports the tax or is male?
Using the method from (a), we get `397/1046+496/1046-160/1046=733/1046~~.701`,
or 237+160+328+8=733 is the size of the event space.
(c) What is the probability that a randomly chosen individual is not unsure or female? `P(notU " or "F)`.
We could proceed as above. However, we need to determine the probability that someone is not unsure. That means the person supports or opposes the tax. So, `P(not U)=397/1046+620/1046=1017/1046`.
So, we get `1017/1046+550/1046-(237+292)/1046=1038/1046~~.992`.
There is an easier approach. The number of people who are unsure that are not female is 8. This is the complement of not unsure or female. So, we take `1-8/1046=1038/1046~~.992`.