thepillow
New member
- Joined
- Sep 12, 2012
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- 34
I was hoping someone could tell me if my work on the question below is correct.
Q: If we roll a normal die \(\displaystyle n\) times, what is the expected number of different sides of the die that we observe?
And here's my attempt at a solution:
Let \(\displaystyle S_i = 1\) if side \(\displaystyle i\) is observed at least 1 time, and \(\displaystyle S_i = 0 \) if side \(\displaystyle i\) is not observed, where \(\displaystyle i = 1, 2, \dots, 6 \). Then for all six possible values of \(\displaystyle i\) we have
since the probability that a side is not observed on each roll is 5/6 and we roll \(\displaystyle n\) times.
And the expected value of each of the six \(\displaystyle S_i\)'s is therefore
Now, let \(\displaystyle N \) be the number of different sides observed. Then we can write \(\displaystyle N \) as
So the expected number of different sides observed is just the expected value of \(\displaystyle N\), which is
Thanks for your comments!
Q: If we roll a normal die \(\displaystyle n\) times, what is the expected number of different sides of the die that we observe?
And here's my attempt at a solution:
Let \(\displaystyle S_i = 1\) if side \(\displaystyle i\) is observed at least 1 time, and \(\displaystyle S_i = 0 \) if side \(\displaystyle i\) is not observed, where \(\displaystyle i = 1, 2, \dots, 6 \). Then for all six possible values of \(\displaystyle i\) we have
\(\displaystyle Pr(S_i = 1) = 1 - Pr(S_i = 0) = 1 - \left(\frac{5}{6}\right)^n \),
since the probability that a side is not observed on each roll is 5/6 and we roll \(\displaystyle n\) times.
And the expected value of each of the six \(\displaystyle S_i\)'s is therefore
\(\displaystyle E(S_i) = 0\cdot Pr(S_i = 0) + 1 \cdot Pr(S_i = 1) = 1 - \left(\frac{5}{6}\right)^n\).
Now, let \(\displaystyle N \) be the number of different sides observed. Then we can write \(\displaystyle N \) as
\(\displaystyle N = S_1 + S_2 + \dots + S_6 \).
So the expected number of different sides observed is just the expected value of \(\displaystyle N\), which is
\(\displaystyle E(N) = E(S_1 + \dots + S_6) = E(S_1) + \dots E(S_6) = 6\cdot E(S_1) = 6\left(1 -\left(\frac{5}{6}\right)^n\right) \).
Thanks for your comments!