Question

A multiple choice has 10 questions. Each question has four answer choices.

a. What is the probability a student randomly guesses the answers and gets exactly six questions correct?
b. Is getting exactly 10 questions correct the same probability as getting exactly zero correct? explain.
c. Describe the steps needed to calculate the probability of getting at least six questions correct if the student randomly guesses.

Answer 1

@rational

Answer 2

for a, is it 1/4*6/10?

Answer 3

@rational

Answer 4

@phi @Callisto @asnaseer

Answer 5

Can you help @phi?

Answer 6

this sounds like you should use binomial distribution
https://en.wikipedia.org/wiki/Binomial_distribution#Probability_mass_function
Have you studied that ?

Answer 7

yes, but I never understood.

Answer 8

There are different levels of understanding. In this case, at least try to write down the formula. can you do that ?

Answer 9

\[P=\left(\begin{matrix}N \\ x\end{matrix}\right)p^x(1-p)^{(N-x)}\]

Answer 10

@phi

Answer 11

@rational can you please come help?

Answer 12

Yes thats the formula
for part a, first figure out the values of N, x, and p

Answer 13

what do they mean? I don't know what they stand for...

Answer 14

N = total number of times an experiment was conducted (10 here)
x = number of successes (6 here)
p = success probability (1/4 here)

Answer 15

ohhh

Answer 16

\[P=\left(\begin{matrix}10 \\ 6\end{matrix}\right)(1/4)^6(1-(1/4))^{(10-6)}\]

Answer 17

right?

Answer 18

Yes evaluate

Answer 19

\[P=\left(\begin{matrix}10 \\ 6\end{matrix}\right)(1/4096)(3/4)^4\]

Answer 20

\[p=\left(\begin{matrix}10 \\ 6\end{matrix}\right)(1/4096)(81/256)\]

Answer 21

\[P=\left(\begin{matrix}10 \\ 6\end{matrix}\right)81/1048576\]

Answer 22

right?

Answer 23

Yes use calculator

Answer 24

to do what?

Answer 25

?...

Answer 26

http://www.wolframalpha.com/input/?i=%2810+choose+6%29*%281%2F4%29^6*%281-1%2F4%29^%2810-6%29

Answer 27

what did I do wrong?

Answer 28

nothing,
your work looks good

Answer 29

just evaluate \(\large \binom{10}{6}\)

Answer 30

how?

Answer 31

it is called binomial coefficient :

\[\large \binom{n}{r} = \dfrac{n!}{r!(n-r)!}\]

Answer 32

you must be familiar with factorials right ?

Answer 33

yes.

Answer 34

okay.

Answer 35

what about b? like I know, but I dunno how to explain it.

Answer 36

@rational

Answer 37

for part b, work both probabilities and see ?

Answer 38

well I know it's different because to get it wrong you have 3/4 but to get it right you have 1/4, but I don't know how to explain it.

Answer 39

nevermind. I got it.

Answer 40

`b. Is getting exactly 10 questions correct the same probability as getting exactly zero correct? explain.`


probability for getting exactly 10 questions correct
\[P(X = 10) ~~~=~~~ \binom{10}{10} (1/4)^{10}(1-1/4)^{10-10}\]


probability for getting exactly 0 questions correct
\[P(X = 0) ~~~=~~~ \binom{10}{0} (1/4)^{0}(1-1/4)^{10-0}\]

Answer 41

do they both evaluate to same number ?

Answer 42

no

Answer 43

thank you... I got that part... now for the hard one, if you're still willing to help, C.

Answer 44

it is not hard as we don't need to do any arithmetic

the question is just asking us to explain how you would work it, its not asking u to actually work it

Answer 45

I guess so.

Answer 46

`c. Describe the steps needed to calculate the probability of getting at least six questions correct if the student randomly guesses.`


so the student should get >= 6 questions correct yes ?

Answer 47

that is he can get 6 or 7 or 8 or 9 or 10 questions correct

so you simply find the probbaility for each like you did in part a and add up

Answer 48

oh, okay... thank you so much.

Answer 49

I really appreciate it.

Answer 50

P(X >=6) := P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10)

Answer 51

yeah, I get it now.

Answer 52

thanks.

Answer 53

yw