# Thread: probability on true/false test

1. ## probability on true/false test

Several students are unprepared for a true/false test with 50 questions, and all of their answers are guesses. Find the mean, variance and standard deviation for the number of correct answers for such students. For mean and variance give 1 decimal place of precision, for standard deviation give 2. Give the values, separated by commas, with 1 blank character between a comma and subsequent value.

2. Originally Posted by gracy
Several students are unprepared for a true/false test with 50 questions, and all of their answers are guesses. Find the mean, variance and standard deviation for the number of correct answers for such students. For mean and variance give 1 decimal place of precision, for standard deviation give 2. Give the values, separated by commas, with 1 blank character between a comma and subsequent value.
This is a question about the binomial distribution. Here the number of correct answers $\displaystyle k$ has distribution $\displaystyle B(k,n,p)$, where [tex]n=50[/math[ the number of Bernouli trials, and $\displaystyle p$ the probability of a favourable outcome on an individual trial.

Facts you should know about the binomial distribution:

$\displaystyle B(k,n,p)={n \choose k}p^k(1-p)^{n-k}$

Mean (or expected) number of favourable outcomes:

$\displaystyle \mu=n\,p$

Variance of the number of favourable outcomes:

$\displaystyle \sigma^2=n\,p\,(1-p)$

and so the standard deviation:

$\displaystyle \sigma=\sqrt{n\,p\,(1-p)}$.

So in this case we have:

$\displaystyle \mu=n\,p=50 \times 0.5=25$

$\displaystyle \sigma^2=n\,p\,(1-p)=50 \times 0.5 \times 0.5=12.5$

$\displaystyle \sigma=\sqrt{n\,p\,(1-p)}=\sqrt{50\times 0.5 \times 0.5}=\sqrt{12.5}\approx 3.54$.

RonL