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A group of $20$ students take the exam. The probability that the student will receive a grade of $2$ - $0.1$, grade $3$ - $0.3$, grade $4$ - $0.4$, grade $5$ - $0.2$. Determine the probability that the group will score more than $80$ points.

Can you tell me in which direction I should think to solve the problem because I have no ideas yet. Thanks in advance!

Mario Grazione
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    This is a very tedious problem. The simplest approach and answer in my opinion is to add the coefficients of $x^{81}$ through $x^{100}$ of the expansion of $(0.1x^2+0.3x^3+0.4x^4+0.2x^5)^{20}$. – JMoravitz Jan 10 '20 at 17:21
  • Use Central Limit Theorm to obtain approximate answer 0,068. – NCh Jan 10 '20 at 17:40

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I actually went onto WolframAlpha, looked up the expansion of $\displaystyle (0.1x^2+0.3x^3+0.4x^4+0.2x^5)^{50}$ and added up the coefficients from $x^{81}$ to $x^{100}$.Beyond $x^{85}$, the contributions reduce 100-fold, so I neglected the rest of the terms for the approximations. I estimated $0.0507 \approx 5.07\%$, though the probability should be slightly higher, but less than $6\%$

Otherwise, we could try simulating this as a normal distribution. This particular distribution of grades has mean $3.7$ and standard deviation $0.9$. For twenty tests, the distribution of total score has mean $3.7\times 20=74$ and standard deviation $0.9 \times \sqrt{20} =4.04$.

Scoring above $80$ approximately corresponds with a z-score above $\displaystyle \frac{80.5-74}{4.04}=1.61$, which occurs with probability $0.0537$, which is a good approximation.

Saketh Malyala
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