Answer:
0.246
Step-by-step explanation:
This question deals with the classic case of determining the exact number of successes (k) in n trials where each trial has only two outcomes and the probability of success in a trial is [tex]p[/tex], probability of failure [tex]= q = 1 - p[/tex]
If we let [tex]P(X = k)[/tex] represent the probability of exactly k successes in n trials, the formula for computing this probability is
[tex]P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}[/tex]
In this experiment there are a total [tex]n = 10[/tex] trials(10 questions total)
[tex]k = 5[/tex] (questions correct)
[tex]p = 0.5[/tex] since you have a 50% chance of randomly getting a T/F question right
[tex]q = 1 - p = 1- 0.5 = 0.5[/tex]
Plugging in the values:
[tex]\[\]\[ P(X = 5) = \binom{10}{5} \times (0.5)^5 \times (0.5)^5 \]\[ P(X = 5) = \frac{10!}{5!(10-5)!} \times 0.5^5 \times 0.5^5 \]\[ P(X = 5) = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} \times 0.5^{10} \]\[ P(X = 5) = 252 \times 0.0009765625 \]\[ P(X = 5) = 0.24609375 \][/tex]
Rounded to the nearest thousandth, the probability of getting exactly half of the 10 true or false questions right by random guessing is 0.246