Improve Test Scores: Solving Probability Problems with Expert Help

  • Thread starter amywilliams99
  • Start date
  • Tags
    Probability
In summary, the student has a 3 in 4 chance of getting 9 questions correct, a 2 in 3 chance of getting 10 questions correct, and a 1 in 3 chance of getting 11 questions correct.
  • #1
amywilliams99
4
0
A student is about to take a test, which consists of 12 multiple choice
questions. Each question has 4 possible answers, of which only one is
correct. The student will answer each question independently. For each
question, there is a probability 2/3 that he knows the correct answer;
otherwise, he picks an answer at random.
(a) Compute the probability that he will get exactly 9 questions right.
(b) compute the probability that his 12th answer will be his 9th correct answer
(c) Compute the probability that his 6th answer will be the 5th correct answer and his 12th answer will be the 9th correct answer.

Please help. For a, I think its binomial distribution, but I am having trouble with the probability, Is it simply 2/3 or must you factor in the probability for guessing. ie - probability is 2/3 +1/4 for the correct answer??
 
Physics news on Phys.org
  • #2
On any given question the probability that he gets the correct answer is 2/3 (knows) + 1/12 (doesn't know and guesses right) = 3/4. Use binomial to work out the probabilities.
 
  • #3
mathman said:
On any given question the probability that he gets the correct answer is 2/3 (knows) + 1/12 (doesn't know and guesses right) = 3/4. Use binomial to work out the probabilities.

Can I ask why 1/12 for the probability of doesn't know and guesses right? Wouldnt it be 1/4, given that there are 4 possible multiple choice answers?
 
  • #4
amywilliams99 said:
A student is about to take a test, which consists of 12 multiple choice
questions. Each question has 4 possible answers, of which only one is
correct. The student will answer each question independently. For each
question, there is a probability 2/3 that he knows the correct answer;
otherwise, he picks an answer at random.
(a) Compute the probability that he will get exactly 9 questions right.

Please help. For a, I think its binomial distribution, but I am having trouble with the probability, Is it simply 2/3 or must you factor in the probability for guessing. ie - probability is 2/3 +1/4 for the correct answer??

Since answering each question is independent from the others, it is sufficient to calculate the probability p that the student will answer the question right. Then the required probability is given by the Bernoulli distribution:

[tex]
P_{9}(12, p) = \left( \begin{array}{c}12 \\ 9 \end{array} \right) p^{9} q^{3}, \ p + q = 1
[/tex]

On to calculating p! This is solved by the formula for total probability under conditional probability. Namely, let event [tex]A[/tex] be: "The student knows the correct answer" and let [tex]\overline{A}[/tex] be: "The student does not know the correct answer". From what is given, we have:

[tex]
P(A) = \frac{2}{3}, \ P(\overline{A}) = 1 - P(A) = \frac{1}{3}
[/tex]

Then, we have [tex]B[/tex] be: "The student chooses the correct answer". We need the conditional probabilities [tex]P(B | A)[/tex] and [tex]P(B | \overline{A})[/tex] to calculate [tex]P(B) \equiv p[/tex]. These are derived using logic and what is given:


[tex]
P(B | A) = 1
[/tex]

because the student will definitely choose the correct answer if he knows the correct answer (event [tex]A[/tex]).


[tex]
P( B | \overline{A}) = \frac{1}{4}
[/tex]

because the student chooses at random out of 4 choices when he does not know the correct answer (event [tex]\overline{A}[/tex]). Then, by the formula for the total probability:


[tex]
p \equiv P(B) = P(A) P(B | A) + P(\overline{A}) P(B | \overline{A}) = \frac{2}{3} \cdot 1 +
\frac{1}{3} \cdot \frac{1}{4} = \frac{2}{3} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4}
[/tex]

All you need to do is substitute in the other numbers and compute the result. Good luck! :)
 
  • #5
Thank you SO much, I understand!
 

What is a probability problem?

A probability problem is a type of mathematical problem that involves calculating the likelihood or chance of a certain event occurring. It is often used in statistics and decision-making to determine the possible outcomes of a situation.

How do I solve a probability problem?

To solve a probability problem, you need to identify the possible outcomes and calculate the probability of each outcome occurring. This can be done using mathematical formulas, tables, or graphs. It is important to accurately define the sample space and use the correct probability rules to arrive at the correct answer.

What are the different types of probability?

There are three main types of probability: theoretical, experimental, and subjective. Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability is based on actual data collected from experiments or observations. Subjective probability is based on personal judgment or opinions.

What is the difference between independent and dependent events?

An independent event is one where the outcome of one event does not affect the outcome of another event. A dependent event is one where the outcome of one event does depend on the outcome of another event. In probability problems, it is important to determine whether events are independent or dependent in order to accurately calculate the probability.

Why is understanding probability important?

Understanding probability is important for a variety of reasons. It allows us to make informed decisions based on the likelihood of certain outcomes. It is also used in fields such as statistics, finance, and science to analyze data and make predictions. In everyday life, understanding probability can help us make better choices and manage risk effectively.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
909
  • Set Theory, Logic, Probability, Statistics
2
Replies
57
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
189
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
992
  • Set Theory, Logic, Probability, Statistics
2
Replies
41
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
9
Views
416
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
2
Replies
45
Views
3K
Back
Top