Let’s try another type of probability problems.
There is a drawer, containing 3 single white socks and 4 single red socks, now you are given three times to draw from the drawer, what’s the probability that you will get a pair of white socks?
To get the answer, always break the problem down to two parts: the numerator, to find all the possibilities where your aimed/interested scenarios happen; the denominator is the entire number of possibilities regardless if anything of your interest transpires or not.
Now let’s look at the easy part – denominator first. The total possibilities are 7 multiplied by 6 = 42. Note we consider the sequence matters on both numerator and denominator. So a scenario such as W W R is counted several times.
Then, to find the possibility where a pair of white socks show up, we can further divide it into two broad scenarios:
- the very first one is W, then if the second is W, conditions satisfied: 3 * 2 *5 =30; if the second is not R, to satisfy the condition, then the third one is W: 3 * 4 * 2 = 24
- the very first one is R, then to satisfy the condition, we have to pick both the second and third W: 4 * 3 * 2 =24.
Adding these up the numerator = 24+24+ 30 = 78. So the final answer = 78/210 = 37.2%.
If we tweak the problem a little bit – say, the three white socks and four red socks are put in one drawer, while in another drawer, there are four red socks and three red ones. You try twice. What’s the probability you get one pair of white socks.
And to make it further complicated, say, there is the third drawer, and you draw twice… You can create a lot of problems yourself and take a stab to solve these made-up questions.
Some people challenge me why you are thinking about these questions, they are supposed to be done in middle school math classes and forgotten now. Well, I recall at that time, the teachers are not the greatest that they just instilled the correct answer without full guidance on how to approach and tackle these interesting problems. Actually, the most important part is the thinking process…
If one can apply this logical reasoning in solving a probability problem, she/he can be empowered to apply the same lucid, clear thinking in tackling lots of problems in life including relationships, economy, politics…