| I had a great deal of trouble understanding some of the concepts of probability.
I struggled to relate some of the questions to real life situations.
This made it difficult to understand. Which in turn made it difficult to answer.
Questions about the probability of A being 0.2, P(B) being 0.4 and P(A|B) = 0.2 left me cold.
I spoke with my dad about this (not because he is good at maths, but because he is good with English)
He suggested that I imagine 100 people in a room.
That some were women and some were men.
And that some were married and some were single.
Then choose a person at random.
The probabilty of A would be the probabilty of the chosen person being a woman
The probability of B would be the probability of the chosen person being married.
Now, whenever I am asked about P(A) and P(B) I have this picture in my mind - it helps me a lot
Some of the following explanations use this approach to understand other probability related things. | | So if P(A) = 0.2 it means that 20 of the 100 people in the room are women
If P(B) = 0.4 it means that 40 people in the room are married
If P(AnB) = 0.08 it means that 8 of the people in the room are women AND are married
P(AUB) is the union of the two groups A and B. In other words The total count of women plus married people.
we know that P(AUB) = P(A) + P(B) - P(AnB)
In other words - The Union of A and B is equal to the total women (20) + the total Marrieds (40) minus those that are both (8)
So if I asked all the women and married people to leave the room, 52 would leave.
| Before I went any further I realised I needed to clearly understand "dependence" and "independence"
If all the people in the room are "normal" then they are independent.
But what do I mean by "normal"
It really means that the relationship between being women (or men)
and being married (or single) is not affected by the selected group.
For example if it were a meeting for unmarried mothers and their parents it would not be a normal distribution.
Independence means that if 20% of the group are women then
20% of them are women regardless of whether they are married or not.
So 20% of the group are women, 20% of the marrieds are women and 20% of the singles are women.
If we then look at something like P(A|B) - the probability of A given that B has already occured - we now know ..
That if P(A) is independent - then the probability of being a woman given that the person is married P(A|B) is ..
the SAME as P(A) - being a woman.
or looking at it another way, if P(A) = P(A|B) then they are independent.
| One of the fairly common questions that I struggled to understand was something similar to the following ...
"If P(A) = 0.2, P(B) = 0.4 and P(AnB) = 0.08
What is the probability that EXACTLY ONE of A and B will occur"
After some discussion I concluded that it was actually asking :
"What is the probability of getting someone who is a woman or married but NOT both?"
so I needed P(AUB) - P(AnB)
The Union of the two groups, but this would include women who were married, so I needed to deduct these.
I already knew that P(AUB) = P(A) + P(B) - P(AnB)
so I ended with P(A) + P(B) - P(AnB) - P(AnB)
or if you prefer P(A) + P(B) - 2P(AnB)
| | More to follow |
| notes
"Probability" is a big word - so I shorten it to "Prob"
that should not be a problem!
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