# Math Help - True or False Probability

1. ## True or False Probability

Wheeewe; this wears me out......
A test consists of 10 true/false questions. To pass the test a student must get 60% or better on the test (Answer 6 or more questions correctly). If a student randomly guesses, what is the probability that the student will pass the test?

2. Hello, Allenge!

A test consists of 10 true/false questions.
To pass the test a student must get 60% or better on the test.
If a student randomly guesses, what is the probability that he/she will pass the test?

The student must get: .6 right or 7 right or 8 right or 9 right or 10 right.

We have: . $P(\text{Right}) = \tfrac{1}{2},\;P(\text{Wrong}) = \tfrac{1}{2}$

We work out the separate probabilities . . .

. $\begin{array}{cccc}\text{6R, 4W} & {10\choose6}(\frac{1}{2})^6(\frac{1}{2})^4 &=& \frac{210}{1024} \\ \\[-4mm]

\text{7R, 3W} & {10\choose7}(\frac{1}{2})^7(\frac{1}{2})^3 &=& \frac{120}{1024} \\ \\[-4mm]

\text{8R, 2W} & {10\choose8}(\frac{1}{2})^8(\frac{1}{2})^2 &=& \frac{45}{1024} \\ \\[-4mm]

\text{9R, 1W} & {10\choose9}(\frac{1}{2})^9(\frac{1}{2})^1 &=& \frac{10}{1024} \end{array}$

$\begin{array}{cccccc}\text{10R, 0W} & {10\choose10}(\frac{1}{2})^{10}(\frac{1}{2})^0 &=& \frac{1}{1024}
\end{array}$

and add: . $P(\text{pass}) \:=\:\tfrac{210}{1024} + \tfrac{120}{1024} + \tfrac{45}{1024} + \tfrac{10}{1024} + \tfrac{1}{1024} \;=\;\tfrac{386}{1024} \:=\:\frac{193}{512}

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