Question

multiple-choice exam a student takes a 14-question, multiple-choice exam with two choices for each question and guesses on each question. find the probability of guessing at least 10 out of 14 correctly. assume the variable is binomial. round the intermediate and final answers to three decimal places.

p (guessing at least 10 out of 14 correctly) = \square

Explanation:

Step1: Identify binomial parameters

$n=14$, $p=0.5$, $P(X\geq10)=P(10)+P(11)+P(12)+P(13)+P(14)$
Binomial formula: $P(k)=\binom{n}{k}p^k(1-p)^{n-k}$

Step2: Calculate $P(10)$

$\binom{14}{10}=\frac{14!}{10!4!}=1001$, $P(10)=1001*(0.5)^{14}\approx0.061$

Step3: Calculate $P(11)$

$\binom{14}{11}=364$, $P(11)=364*(0.5)^{14}\approx0.022$

Step4: Calculate $P(12)$

$\binom{14}{12}=91$, $P(12)=91*(0.5)^{14}\approx0.005$

Step5: Calculate $P(13)$

$\binom{14}{13}=14$, $P(13)=14*(0.5)^{14}\approx0.001$

Step6: Calculate $P(14)$

$\binom{14}{14}=1$, $P(14)=1*(0.5)^{14}\approx0.000$

Step7: Sum probabilities

$0.061+0.022+0.005+0.001+0.000=0.089$

Answer:

0.089