Question

A test consists of 10 true or false questions. To pass the test a student must answer at least eight questions correctly. If the student guesses on each question, what is the probability that the student will pass the test

Answer 1

The probability of any student passing the given test is 0.0538.

Step-by-step explanation:

Here, the total number of questions in the test   = 10

The minimum correct answers needed to pass the test  = 8 or more

⇒  n ≥ 8

Now, since there are only 2 options for each question:

So, the probability of answering it right   =  $(\frac{1}{2} ) = 0.5 = p$

And the probability of answering it wrong    =  $(\frac{1}{2} ) = 0.5 = q$

Now, let us use the Binomial Distribution Formula here:

$P(x) = ^nC_x p^xq^{(n-x)}$

Now, solving for x = 8 , 9 and 10

When x  = 8 , P(8) is given as :

$P(8) = ^{10}C_8 (0.5)^8(0.5)^{(10-8)} = 45\times (0.5)^8(0.5)^2 = 0.0439$

When x  = 9 , P(9) is given as :

$P(9) = ^{10}C_9 (0.5)^9(0.5)^{(10-9)} = 10\times (0.5)^9(0.5)^1 = 0.009$

When x  = 10 , P(10) is given as :

$P(10) = ^{10}C_{10} (0.5)^10(0.5)^{(10-10)} = 10\times (0.5)^{10}(0.5)^0 = 0.0009$

Now, adding the 3 probabilities, we get:

(P  ≥ 8) =  P(8) + P(9) + P(10) =  0.0439 + 0.009 + 0.0009 = 0.0538

or, (P  ≥ 8) =  0.0538

Hence,  the probability that the student will pass the test is 0.0538.