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प्रश्न
One mapping (function) is selected at random from all the mappings of the set A = {1, 2, 3, ..., n} into itself. The probability that the mapping selected is one to one is ______.
पर्याय
`1/n^n`
`1/n`
`(n - 1)/(n^(n - 1))`
None of these
उत्तर
One mapping (function) is selected at random from all the mappings of the set A = {1, 2, 3, ..., n} into itself. The probability that the mapping selected is one to one is `(n - 1)/(n^(n - 1))`.
Explanation:
Total number of mappings from a set A having n elements onto itself is nn
Now, for one-to-one mapping the first element in A can have any of the n images in A; the 2nd element in A can have any of the remaining (n – 1) images, counting like this, the nth element in A can have only 1 image.
Therefore, the total number of one-to-one mappings is n.
Hence the required probability is `n/n^n = (n(n - 1))/(n n^(n - 1)) = (n - 1)/(n^(n - 1))`.
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Which of the following can not be valid assignment of probabilities for outcomes of sample space S = {ω1, ω2,ω3,ω4,ω5,ω6,ω7}
| Assignment | ω1 | ω2 | ω3 | ω4 | ω5 | ω6 | ω7 |
| (a) | 0.1 | 0.01 | 0.05 | 0.03 | 0.01 | 0.2 | 0.6 |
| (b) | `1/7` | `1/7` | `1/7` | `1/7` | `1/7` | `1/7` | `1/7` |
| (c) | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 |
| (d) | –0.1 | 0.2 | 0.3 | 0.4 | -0.2 | 0.1 | 0.3 |
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