Question

12) A student takes a 10-question multiple-choice test by guessing. Each question has 4 choices. Use the binomial distribution to compute the following probabilities. a. b. c. The student gets at most 2 correct. The student gets exactly 3 correct. The student gets at least 7 correct.

Answer 1

12)here for binomial distribution parameter n=10 and p=1/4 (probability of getting a question correct)

a)P(X<=2) =P(X=0)+P(X=1)+P(X=2)=¹⁰C₀(1/4)⁰(3/4)¹⁰+¹⁰C₁(1/4)¹(3/4)⁹+¹⁰C₂(1/4)²(3/4)⁸=0.5256

b)P(X=3)=+¹⁰C₃(1/4)³(3/4)⁷ =0.2503

c)

P(X>=7)=P(X=7)+P(X=8)+P(X=9)+P(X=10)

=¹⁰C₇(1/4)⁷(3/4)³+¹⁰C₈(1/4)⁸(3/4)²+¹⁰C₉(1/4)⁹(3/4)¹ +¹⁰C₁₀(1/4)¹⁰(3/4)⁰ =0.0035

10)

x	f(x)	xP(x)	x2P(x)
1	1/3	0.333	0.333
2	1/8	0.250	0.500
3	1/8	0.375	1.125
4	1/4	1.000	4.000
5	1/6	0.833	4.167
	total	2.792	10.125
	E(x) =μ=	ΣxP(x) =	2.7917
	E(x2) =	Σx2P(x) =	10.1250
	Var(x)=σ2 =	E(x2)-(E(x))2=	2.3316
	std deviation=	σ= √σ2 =	1.5270

from above mean =2.7917

variance= 2.3316

std deviation=1.5270