**A**: This is similar to the coin flipping problem (i.e what is the probability of obtaining exactly 4 heads if a fair coin is flipped 5 times) except in this case the probably is not 1/2.

- to get at least 20 points, one must get at least 4 questions right. so we are interested in P(X>=4)
- The random variable X is binomial with n = 5 and p = 0.25 (4 choices)

- P(X>=4) = P(4) + P(5) = 0.0155.
- P(4) = C(4, 5) * p^4 * (1-p)^(5-4) = {5!/[(5 - 4)! x 4!]} * 0.25^4 * (1-0.25)^(5-4) = 0.0146. Or with Excel,
- P(4) = ( fact(5)/(fact(5-4)*fact(4)) ) * 0.25^4 * (1-0.25)^(5-4)
- P(4) = BINOMDIST(4,5,0.25,FALSE)

- P(5) = BINOMDIST(5,5,0.25,FALSE) = 0.0009

- P(4) = C(4, 5) * p^4 * (1-p)^(5-4) = {5!/[(5 - 4)! x 4!]} * 0.25^4 * (1-0.25)^(5-4) = 0.0146. Or with Excel,

*Bonus Points*

- For a binomial random variable, the probability, p, of success must be constant for all trials, and the trials are independent.

*Category: C++ Quant > Quantitative Analysis > Probability*

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