By J.V. Uspensky
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Extra resources for Introduction to mathematical probability
The required probability can be ability of drawing at least one white ball. expressed in two ways. 1 First expression: b(b - 1) _ (a + b)(a + b - • • 1) • • (b - n + 1) • • (a + b + 1) n - Second expression: a a + b [ 1 + b a + b + + 1 - · · · b(b - 1) • • (b · (a + b - 1)(a + b - 2) · - • · + 2) (a + b n - n Equating them, we have an identity 1 + b a + b - 1 + · · · + b(b - 1) • • • (b [ (a + b - 1)(a + b - 2) = a + b a- 1 - · - · · n + 2) (a + b - n + 1) b(b - 1) • • • (b - n + 1) (a + b)(a + b - 1) ] + 1) .
Let us denote by What is the probability Pm the required probability. It is not apparent bow we can find the explicit expression for this probability, but using the theorems of total and compound probability, we can form equations which yield the desired expression for Pm without any difficulty. To this end, let us first find the probability drawn does not exceed m. P that the greatest number It is obvious that this may happen in m mutually exclusive ways; namely, when the greatest number drawn is 1, 2, 3, and so on up tom.
Solution. To find the number of all possible cases in this problem, suppose that we distinguish the balls by numbering them from 1 to n. The ball with the number 1 may fall into any of theN compartments, which givesN cases. The ball with the number 2 may also fall into any one of theN compartments; so that the number of cases for 2 balls will beN·N = N2• Likewise, for 3 balls the number of cases will be N2 and for any number n • N = N3, of balls the number of cases will be N". To find the number of favorable cases, first suppose that a group of h specified balls falls into a designated compartment.
Introduction to mathematical probability by J.V. Uspensky