Respuesta :
To solve the following problem, we must use the binomial probability equation. This is expressed mathematically as:
P (r) = [n!/(n-r)! r!] * p^r * q^(n-r)
where,
n = total number of questions = 4
r = number of correct questions = 4
p = probability of success = 25% = 0.25
q = probability of failure = 0.75
Therefore substituting the values into the given equation will give us:
P (r = 4) = [4! / 0! 4!] * 0.25^4 * 0.75^0
P (r = 4) = 3.906 * 10^-3 = 0.391%
Answer: Richard only has 0.391% to answer all questions correctly by guessing.
The probability of answering all the question correctly is [tex]\boxed{\bf \dfrac{1}{256}}[/tex]
Further Explanation:
Given:
The total number of question is [tex]4[/tex].
Each question has [tex]4[/tex] options and one is correct.
Concept used:
Probability is defined as the ratio of number of favorable outcomes to number of total outcomes.
[tex]\boxed{\text{Probability}=\dfrac{\text{favorable outcomes}}{\text{total outcomes}}}[/tex]
Calculation:
The probability of getting the correct answer is calculated as follows:
[tex]\begin{aligned}P&=\dfrac{1}{4}\\&=0.25\end{aligned}[/tex]
The probability of getting the incorrect answer is calculated as follows:
[tex]\begin{aligned}P&=1-\dfrac{1}{4}\\&=1-0.25\\&=0.75\end{aligned}[/tex]
The formula for probability of answering all question correctly is as follows:
[tex]\boxed{P'=^nC_{r}P^{r}(1-P)^{n-r}}[/tex]
Here, [tex]n[/tex] is number of question and [tex]r[/tex] is number of correct answer, [tex]P'[/tex] is the probability of answering all question correctly.
Solve the above equation to obtain probability of answer all question correctly.
Substitute [tex]4[/tex] for [tex]n[/tex], [tex]0.25[/tex] for [tex]P[/tex] and [tex]4[/tex] for [tex]r[/tex] in above equation to obtain probability of answering all question correctly.
[tex]\begin{aligned}P'&=^4C_{4}(0.25)^{4}(1-0.25)^{4-4}\\&=\dfrac{4!}{(4-4)!\cdot 4!}\left(\dfrac{25}{100}\right)^{4}(0.75)^{0}\\&=\left(\dfrac{1}{4}\right)^{4}\\&=\dfrac{1}{256}\end{aligned}[/tex]
Therefore, the probability of the correct answer is [tex]0.25[/tex].
The probability of the incorrect answer is [tex]0.75[/tex].
The probability of answering all question correctly is [tex]\dfrac{1}{256}[/tex].
Thus, the probability of answering all question correctly is [tex]\boxed{\dfrac{1}{256}}[/tex].
Learn more:
1. Simplification: https://brainly.com/question/1602237
2. Quadratic equation: https://brainly.com/question/1332667
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Probability
Keywords: Probability, Question, Richard, multiple choice, four answer, class, attend, guesses, correctly answer, favorable outcomes, total outcomes.
