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  1. # 5.2
  2.  
  3. ## a
  4.  
  5. The probability that the first observation is not he jth from the original sample is $\frac{n-1}{n}$. There are $n-1$ options that aren't the jth from the original sample, and n total options.
  6.  
  7. ## b
  8.  
  9. The same as above($\frac{n-1}{n}$), as bootstrapping samples without replacement.
  10.  
  11. ## c
  12.  
  13. Probability an individual sample is not the jth observation: $\frac{n-1}{n}=(1-\frac{1}{n})$.
  14.  
  15. Total number of samples: $n$.
  16.  
  17. Thus, the probability that all samples aren't the jth is $(1-1/n)^n$.
  18.  
  19. ## d
  20.  
  21. $(1-1/5)^5=(4/5)^5=0.32768$
  22.  
  23. ## e
  24.  
  25. $(1-1/100)^{100}=0.366$
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