Answer:
(a)0.208
(b)0.633
Explanation:
Since the variable X is not defined, we assume that X is the number of questions answered correctly.
Each question has 4 options out of which just 1 is correct.
Each question is independent of other questions. Thus, we can use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]\begin{gathered} P(X=x)=^nC_x(p^x)(1-p)^{n-x} \\ where\; ^nC_x=\frac{n!}{(n-x)!x!} \end{gathered}[/tex]
• In this case, there are 8 questions: n=8
,
• 1 out of 4 options is correct, p=1/4=0.25
Part 1
[tex]\begin{gathered} P(3)=^8C_3(0.25^3)(1-0.25)^{8-3} \\ =56\times0.25^3\times0.75^5 \\ =0.208 \end{gathered}[/tex]
P(3)=0.208 correct to 3 decimal places.
Part 2
[tex]P(X>1)=1-P(X\le1)[/tex]
First, we calculate P(X≤1):
[tex]\begin{gathered} P\mleft(X\le1\mright)=P(0)+P(1) \\ P(0)=^8C_0(0.25^0)(0.75)^8=1\times1\times0.1001=0.1001 \\ P(1)=^8C_1(0.25^1)(0.75)^7=8\times0.25\times0.1335=0.2670 \\ \implies P(X\le1)=0.1001+0.2670=0.3671 \end{gathered}[/tex]
Therefore, the probability, P(more than 1) is:
[tex]P(X>1)=1-P(X\le1)=1-0.3671=0.633[/tex]
P(more than 1) is 0.633 (correct to 3 decimal places).
