# Calculating binomial probability question

In a ten-question true-false exam, find the probability that a student gets a grade of 70 percent or better by guessing. Answer the same question if the test has 30 questions, and if the test has 50 questions.

My reasoning is the following:

We are looking for a binomial probability where $n=10$, success$=7$ failure $=3$

then the solution should be: $$10 \cdot 7^{0.5} \cdot 3^{0.5} = 45.82$$

for the other two questions:

$$30\cdot 21^{0.5} \cdot 9^{0.5}$$

and for $50$: $$50\cdot 35^{0.5}\cdot 15^{0.5}$$

IS this correct?

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It could not be right, probability is always a number between $0$ and $1$. In the case of $10$ questions, it is exactly the same as for the following problem. Toss a fair coin $10$ times. What is the probability of $7$ or more heads? –  André Nicolas Oct 20 '11 at 11:21
thats 1/2ˆ7. so for the question its just 0.5ˆ7? –  Dbr Oct 20 '11 at 11:40
The probability of exactly $k$ heads is $\binom{10}{k}/2^{10}$, so probability of $7$ or more is $\binom{10}{7}+\binom{10}{8}+\binom{10}{9}+\binom{10}{10}$, all divided by $2^{10}$. For larger numbers the calculation gets messy, one usually uses a normal approximation to the binomial. –  André Nicolas Oct 20 '11 at 11:54
There is a discussion on Wikipedia. For the normal approximation, go to Section 5.4. Most basic probability books cover this. A quick web search would find more detailed worked examples. Basically if the number of T/F questions is $n$, and $n$ is not too small, the binomial distribution is well-approximated by the normal with mean $n/2$ and variance $n/4$. In your problem $n=10$ (too small), $n=30$ (marginal), $n=50$ (probably OK). –  André Nicolas Oct 20 '11 at 14:11
Is there a difference between $7^{0.5}\cdot 3^{0.5}$ and $(0.5)^{7}\cdot (0.5)^{3}$? Dbr seems to be using the strangest calculation I have ever seen, and getting a probability value of $45.82$ doesn't faze him (her?) the least bit even though @Andre pointed it out to him barely minutes after the question was posted about $9$ hours ago! –  Dilip Sarwate Oct 20 '11 at 20:36