Respuesta :
Answer:
(a) The probability of guessing 20 answers correctly is 1.15*10^-8 or 0.000001%
(b) The probability of answer fewer than 5 questions correctly is 0.2138 or 21.38%.
Step-by-step explanation:
If the student is guessing, he or she has only 1 out of 4 chances of answer correctly. This is equivalent to a probability of success p=0.25.
For this problem we can use the binomial distribution to calculate the probabilities.
(a) What is the probability that the student answers more than 20 questions correctly?
We have a total of 25 questions. The probability of guessing 20 answers correctly is
[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}\\\\P(20)=\frac{25!}{20!5!}*0.25^{20}*0.75^5\\\\P(20)=53130* 9.095*10^{-3}*0.2373\\\\P(20)=1.15*10^{-8}[/tex]
The probability of guessing 20 answers correctly is 1.15*10^-8 or 0.000001% .
(b) The probability of answer fewer than 5 questions correctly is the sum of:
1) Answering 0 questions correctly
2) Answering 1 questions correctly
3) Answering 2 questions correctly
4) Answering 3 questions correctly
5) Answering 4 questions correctly
This probabilities can be calculated as
[tex]P(0)=\frac{25!}{0!25!}*0.25^0*0.75^{25}=1*1*7.53*10^{-4}=0.0008\\\\P(1)=\frac{25!}{1!24!}*0.25^1*0.75^{24}= 0.0063
\\\\P(2)=\frac{25!}{2!23!}*0.25^2*0.75^{23}= 0.0251
\\\\P(3)=\frac{25!}{3!22!}*0.25^3*0.75^{22}= 0.0641
\\\\P(4)=\frac{25!}{4!21!}*0.25^4*0.75^{21}= 0.1175
[/tex]
Then the sum of the probabilty of this events is
[tex]P(x<5)=P(0)+P(1)+P(2)+P(3)+P(4)[/tex]
[tex]P(x<5)=0.0008+0.0063+0.0251+0.0641+0.1175=0.2138[/tex]
The probability of answer fewer than 5 questions correctly is 0.2138 or 21.38%.
