Some people didn't understand my post about why induction is self-refuting. I believe this may be because they don't know exactly how the odds of something are calculated. By way of illustration, we shall imagine that a person takes a cancer test, which turns out positive.
We know that the cancer test has a 95 percent confidence level - that is that it generates false positives and false negatives only 5 percent of the time. Some people might think because of this that the person has a 95 percent chance of having cancer. That is not true. As a matter of fact, there is not enough information in the problem to know what the person's chance of having cancer is.
Let us assume that the person has an 8 percent base chance of having cancer. Perhaps this is merely an informed guess by the doctor or perhaps it comes from the percentage of people in the general population who have cancer. As such, anyone walking in the door might be expected to have cancer based on their age, their symptoms, or whatever. This is the piece of information we will need to calculate the chance of having cancer.
So if 8 percent of the population has cancer then out of every 10,000 people that come to be tested 800 of them will have cancer and 9,200 will not. After the test we will have four groups: 760 people who have cancer and tested positive for cancer (800 x .95) and 40 people who have tested negative for cancer, but really do have it (800 x .05). We will also have 460 people who have tested positive for cancer, but do not have cancer. We will also have 8,740 people who do not have cancer and have tested negative for it.
What then is chance that the person really does have cancer after receiving a positive test result for cancer? The answer is 760 / (760+460) ... that is, the number of people who have cancer and have tested positive, divided by the size of the whole number of people who have tested positive for cancer (the 760 plus the 460 who falsely tested positive). Accordingly we can say that the person has a 62.3 percent chance of really having cancer.
It may seem strange to you that a test that is 95 percent accurate results only in a 62.3 percent chance of the person having cancer. However, you need to look at it this way: The person's initial chance of having cancer (8 percent) has now increased to 62.3 percent and a second positive test would let us increase the probability accordingly. The more tests we do, the greater the chance of the person having cancer if, indeed, they do have cancer.
We know that the cancer test has a 95 percent confidence level - that is that it generates false positives and false negatives only 5 percent of the time. Some people might think because of this that the person has a 95 percent chance of having cancer. That is not true. As a matter of fact, there is not enough information in the problem to know what the person's chance of having cancer is.
Let us assume that the person has an 8 percent base chance of having cancer. Perhaps this is merely an informed guess by the doctor or perhaps it comes from the percentage of people in the general population who have cancer. As such, anyone walking in the door might be expected to have cancer based on their age, their symptoms, or whatever. This is the piece of information we will need to calculate the chance of having cancer.
So if 8 percent of the population has cancer then out of every 10,000 people that come to be tested 800 of them will have cancer and 9,200 will not. After the test we will have four groups: 760 people who have cancer and tested positive for cancer (800 x .95) and 40 people who have tested negative for cancer, but really do have it (800 x .05). We will also have 460 people who have tested positive for cancer, but do not have cancer. We will also have 8,740 people who do not have cancer and have tested negative for it.
What then is chance that the person really does have cancer after receiving a positive test result for cancer? The answer is 760 / (760+460) ... that is, the number of people who have cancer and have tested positive, divided by the size of the whole number of people who have tested positive for cancer (the 760 plus the 460 who falsely tested positive). Accordingly we can say that the person has a 62.3 percent chance of really having cancer.
It may seem strange to you that a test that is 95 percent accurate results only in a 62.3 percent chance of the person having cancer. However, you need to look at it this way: The person's initial chance of having cancer (8 percent) has now increased to 62.3 percent and a second positive test would let us increase the probability accordingly. The more tests we do, the greater the chance of the person having cancer if, indeed, they do have cancer.
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