1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
lubasha [3.4K]
3 years ago
15

In a multiple choice quiz there are 5 questions and 4 choices for each question (a, b, c, d). Robin has not studied for the quiz

at all, and decides to randomly guess the answers. Find the probabilities of each of the following events:
a. The first question she gets right is the 3rd question?
b. She gets exactly 3 or exactly 4 questions right?
c. She gets the majority of the questions right?
Mathematics
1 answer:
Flauer [41]3 years ago
6 0

Answer:

a \mathbf{P(X=3)=0.1406}

b \mathbf{\[P\left( {X = 3 \ or \ 4 } \right) =  0.1025}

c  \mathbf{\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) =  0.1035}

Step-by-step explanation:

Given that:

In a multiple choice quiz:

there are 5 questions

and 4 choices for each question (a, b, c, d)

let X be the correctly answered question = 1 answer only

and Y be the choices for each question = 4 choices

The probability that Robin guessed the correct answer is:

Probability = n(X)/n(Y)

Probability = 1//4

Probability = 0.25

The probability mass function is :

P(X=x)=0.25 (1-0.25)^{x-1}

We are to find the required probability that the first question she gets right is the 3rd question.

i.e when x = 3

P(X=3)=0.25 (1-0.25)^{3-1}

P(X=3)=0.25 (0.75)^{2}

\mathbf{P(X=3)=0.1406}

b) Find the probability that  She gets exactly 3 or exactly 4 questions right

we know that :

n = 5 questions

Probability P =0.25

Let represent X to be the number of questions guessed correctly i,e 3 or 4

Then; the probability mass function can be written as:

\[P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\x\\\end{array}} \right){\left( {0.25} \right)^x}{\left( {1 - 0.25} \right)^{5 - x}}\]

P(X = 3 \ or  \ 4)= P(X =3) +P(X =4)

\[P\left( {X = 3 \ or \ 4 } \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\3\\\end{array}} \right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] +  \left( {\begin{array}{*{20}{c}}\\5\\\\4\\\end{array}} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\]

\[P\left( {X = 3 \ or \ 4 } \right) =  \dfrac{5!}{3!(5-3)!}\right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] + \dfrac{5!}{4!(5-4)!} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\]

\[P\left( {X = 3 \ or \ 4 } \right) =  \dfrac{5!}{3!(2)!}\right){\left( {0.25} \right)^3}{\left( {0.75} \right)^{2}}\] + \dfrac{5!}{4!(1)!} \right){\left( {0.25} \right)^4}{\left( {0.75} \right)^{1}}\]

\[P\left( {X = 3 \ or \ 4 } \right) =  0.0879+0.0146

\mathbf{\[P\left( {X = 3 \ or \ 4 } \right) =  0.1025}

c) Find the probability if She gets the majority of the questions right.

We know that the probability mass function is :

\[P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\x\\\end{array}} \right){\left( {0.25} \right)^x}{\left( {1 - 0.25} \right)^{5 - x}}\]

So; of She gets majority of her answers right ; we have:

The required probability is,

P(X>2) = P(X=3) +P(X=4) + P(X=5)

∴

\[P\left( {X = 3 \ or \ 4 \ or  \ 5 } \right) = \left( {\begin{array}{*{20}{c}}\\5\\\\3\\\end{array}} \right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] +  \left( {\begin{array}{*{20}{c}}\\5\\\\4\\\end{array}} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\]+  \left( {\begin{array}{*{20}{c}}\\5\\\\5\\\end{array}} \right){\left( {0.25} \right)^5}{\left( {1 - 0.25} \right)^{5 - 5}}\]

\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) =  \dfrac{5!}{3!(5-3)!}\right){\left( {0.25} \right)^3}{\left( {1 - 0.25} \right)^{5 - 3}}\] + \dfrac{5!}{4!(5-4)!} \right){\left( {0.25} \right)^4}{\left( {1 - 0.25} \right)^{5 - 4}}\] + \dfrac{5!}{5!(5-5)!} \right){\left( {0.25} \right)^5}{\left( {1 - 0.25} \right)^{5 - 5}}\]

\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) =  \dfrac{5!}{3!(5-3)!}\right){\left( {0.25} \right)^3}{\left( {1 - 0.75} \right)^{2}}\] + \dfrac{5!}{4!(5-4)!} \right){\left( {0.25} \right)^4}{\left( {0.75} \right)^{1}}\] + \dfrac{5!}{5!(5-5)!} \right){\left( {0.25} \right)^5}{\left( {0.75} \right)^{0}}\]

\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) =  0.0879 + 0.0146 + 0.001

\mathbf{\[P\left( {X = 3 \ or \ 4 \ or \ 5 } \right) =  0.1035}

You might be interested in
Pls help a girl out .......
VARVARA [1.3K]

Answer:

Yes

Step-by-step explanation:

The sentence does correctly state the equation written.

8 0
3 years ago
How far is Mars from the sun actually formatted
pav-90 [236]

Answer:

228 kilometers away

Step-by-step explanation:

6 0
3 years ago
You spent $10.50 at the fair. If it costs $4.50 for admission and you rode
7nadin3 [17]
150 each because he will have $6 left
7 0
3 years ago
Read 2 more answers
Merit works in a florist shop and makes flower arrangements
Vikentia [17]
So what is the question
5 0
3 years ago
What is 2×3(10)^-5 in standard form
Sindrei [870]
2×3(10)^{-5}=6×(10)^-5=\frac{6}{10^5} =0.00006
5 0
3 years ago
Other questions:
  • translate triangle A by vector (-3,1) to give triangle B. Then roste your triangle B 180 degrees around the origin to give trian
    6·1 answer
  • Angle x is (2n+12) and angle y is (3n+18). Find the measure of angle x
    7·1 answer
  • Choose the correct factorization for the polynomial.
    9·1 answer
  • I have to factor this expression (ax^2+bx+c) and I’m having some trouble. Can someone help me with this?
    9·2 answers
  • HELP MEEEEEE!!! This is due today !!!!!
    12·2 answers
  • $14.90 with $0.05 5% sales tax
    8·2 answers
  • Show that (×)× = ×(×) if and only if the vectors and are collinear.
    15·1 answer
  • If h(x)=3x + 4 , what is h(2x-3)
    10·1 answer
  • What's four minus 1 third​
    8·2 answers
  • Which situation can be represented by y=12x?
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!