Respuesta :
Answer:
P(Fewer than 3) = 0.05.
Step-by-step explanation:
We are given that a student takes a true-false test that has 10 questions and guesses randomly at each answer.
The above situation can be represented through Binomial distribution;
[tex]P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....[/tex]
where, n = number of trials (samples) taken = 10 questions
r = number of success = fewer than 3
p = probability of success which in our question is probability
that question is answered correctly, i.e; 50%
LET X = Number of questions answered correctly
So, it means X ~ Binom(n = 10, p = 0.50)
Now, Probability that Fewer than 3 questions are answered correctly is given by = P(X < 3)
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
= [tex]\binom{10}{0}\times 0.50^{0} \times (1-0.50)^{10-0}+ \binom{10}{1}\times 0.50^{1} \times (1-0.50)^{10-1}+ \binom{10}{2}\times 0.50^{2} \times (1-0.50)^{10-2}[/tex]
= [tex]1 \times 0.50^{10} + 10 \times 0.50^{10} +45 \times 0.50^{10}[/tex]
= 0.05
Hence, the P(Fewer than 3) is 0.05.
