# Not sure what i did wrong binomial probability

1. Nov 13, 2011

### mtingt

In a 22-item true–false examination, a student guesses on each question.

If 14 correct answers constitute a passing grade, what is the probability the student will pass?

i did c(22,14)* (1/2)^14 * (1/2)^8

2. Nov 13, 2011

### mathman

You have to add in the probabilities for more than 14 correct guesses. Your answer is for exactly 14 correct.

3. Nov 13, 2011

### mtingt

so i would have to add every single probability up to 22?
is there any other way i could do this?

4. Nov 13, 2011

### SW VandeCarr

You can use an online calculator for p=0.5, n=22, x=14 and solve for $P(X\geq x)$

http://stattrek.com/tables/binomial.aspx

5. Nov 13, 2011

### awkward

You might also use a Normal approximation to the Binomial distribution.

6. Nov 14, 2011

### SW VandeCarr

The normal approximation gives p=0.1004 whereas the presumably exact binomial gives $(P(X\geq x)=0.1431$ for x=14.

For the normal approximation I'm using mean 11 and $SD = \sqrt {11(1-0.5)} = 2.345$

7. Nov 14, 2011

### awkward

I get $P(X \geq 13.5) = 0.1432$ using the Normal distribution adjusted for continuity.

8. Nov 14, 2011

### SW VandeCarr

I did too, but when Ted Williams was told his 0.3995 batting average would go into the record books as 0.400, he said that wasn't really 0.400 and played through two final season games ending up with a 0.406 batting average. Is 13.5 a passing grade or is 14 a threshold value? I understand the continuity correction and it's fine for some applications but for n=22 and a "threshold" value, why not use an exact calculation? In either case, you will likely use tables or a calculator.

Having said that, it's closer than I would have thought, but I wouldn't have been comfortable without doing the exact approach.

Last edited: Nov 14, 2011
9. Nov 14, 2011

### awkward

I agree, in this case the approximation works better than we have any right to expect. Still, it's a useful tool to have around.