Bionomial Distribution

[desh]

In exam there are 10 questions with 5 options, if you will score 3 or more correct you will pass. What is probability that you will pass just by guessing ????


Ans: The passing rate is : 1/5 =0.2, Failing rate=4/5=0.8
Passing rate by guessing 3 or more correct questions =1- (failing by 1 correct question+2correct question+0correct question)

P(0) = 10!/0!10! *(0.2)^0 *0.8^10 = 0.1073
P(1) = 10!/1! 9! (0.2)^1 *.8^9 = 0.2684
P(2) = 10!/2!8! (0.2)^2 @0.8^8 =0 .3020

Probability of Passing by scoring 3 or more correct answers by guessing =1-0.6777 =0.3223
approx: 32%
is this the correct approach or any other method is required

 

[brian.field]

I haven't verified the calculations but the logic is sound.

 

[desh]

thanks @brian.field it boosts confidence .

 

[David Harper CFA FRM]

@desh That's the best (and only?) way I know and my excel matches your calculations. One thing to note is that we can always look for the normal approximation (often the case in a backtest). One test is that n*p > 5, but here 10*20% = 2, so we don't expect a good normal approximation. To do that we just standardize (3 - µ)/σ = (3 - 10*20%)/sqrt(80%*20%*2) = 0.791; i.e., 3 is + 0.791 standard errors above the mean of 2. Then 1 - N(Z = +0.791) = 1 - Pr(Z > 0.791) ~= 21.5%. So normal is a terrible approximation here, but if np>5, this doesn't require itemizing each outcome. Thanks!