Example: 1  
(a) A bag contains red and white balls, all identical in shape and size. A person puts his hand in and without looking, pulls out a ball. What is the probability of it being red?
(b) In the same bag, he puts his hand in and draws out a pair of balls. What is the probability of both of them being red?

Solution: 1(a)  
It should be obvious that drawing a white ball is more likely than drawing a red one, since there are more white balls. Precisely,
so that,

Solution: 1(b)  
There are red balls, so there are pairs of red balls possible. (Some readers might raise an objection here: if the balls are identical, how can there be ways of forming pairs? There should be just one way, as discussed in &. These readers should try to appreciate the difference between number of pairs and number of ways of forming pairs. The former is , the latter is .
The total number of pairs possible is . Thus
Similarly, we will have
Note that

Example: 2  
There are two bags, one containing white and black balls, and the other containing white and black balls. A person draws one ball at random from each bag. Find the probability that

Solution: 2(a)  
Let the two bags be labelled and respectively. Note that drawing a ball from is independent of drawing a ball from . Thus,

Solution: 2(b)  
We note that the case {one white and one black} is possible in two ways:
Note that and are mutually exclusive events.
Thus,
We could also have calculated this probability by noting that
and since , we have

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