The probability that certain events X and Y occur at the same time is called the joint probability.
The shaded area for all events (blue frame) is as shown below. ∩ is called a cap.
■example
What is the probability of rolling two dice at the same time and getting a 1 for both?
■answer
■What is Conditional probability?
The probability that another event will occur on the premise that one event will occur is called the conditional probability.
For example, the probability that X will occur on the assumption that Y has occurred is expressed as follows. The shaded area for the Y event (blue frame). "|" Is called a vertical bar.
Read "probability of X given Y".
■example
What is the probability that the sum of the dice rolled twice will be 9 or more? However, the first roll is 5.
■answer
In the above example, you may not feel the need for this formula because you can intuitively understand that the answer is just to find the probability P (X) that the second dice will be 4 or more.
But what about the following cases?
■example
It turns out that one couple has two children, and at least one of the two is a boy. What is the probability that they are both boys at this time?
However, the probability that a man and a woman will be born is half and half.
■answer
From the above, it is easier to understand if you use the formula this time.
Furthermore, it is a good example to use the formula even in the following cases.
■A famous example
There is a disease that affects 1% of the population. When testing for this disease, there is a 10% chance that it will be falsely positive despite being normal.
There is a 10% chance that you will be misjudged as negative despite your illness. If a person here is positive, what is the probability of being really ill?
■answer
it's better to understand in the figure.
It will be as above. Let's actually calculate.
In other words, even if it is judged to be positive, there is only a 8% chance that you are really sick.
If you're positive, you're 90% likely to be really ill, so you might think you're more likely to be ill, but there are actually more people who aren't ill but are positive.
Therefore, the ratio is low.
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