True/false examinations by tossing a coin

  • Thread starter Cyrus
  • Start date
In summary, the conversation discusses a student who takes a true/false exam and must answer at least 70% correctly to pass. The probability of passing is calculated for different numbers of questions, with the results being 5.5%, 2.1%, 0.13%, and 0 for 10, 20, 50, and 100 questions respectively. The conversation also raises a question about the probability of getting caught cheating and asks for the probability of being caught over the course of 2, 5, and 10 years.
  • #1
Cyrus
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2.) A student selects his answers on a true/false examinations by tossing a coin (so that any particular answer has a 0.50 probability of being correct). He must answer at least 70% in order to pass. Find the probability of passing when the number of questions is:
(a) 10 (b) 20 (c) 50 (d) 100

EDIT: None of that multiplication rule crap should apply here, D'OH!

I think this too is another case of the binomial probability; hence:

[tex] X \sim Bin(10,0.5) [/tex]

I must evaluate the cases where,

n=10,20,50,100

Ah! The 70% comes into play for the probablity. I want probablity of greater than 70%, or for values of [tex] X \geq .7n [/tex]

Problem a.)

From the table,

[tex] P(X \geq 7) = 1- 0.945 [/tex]

So,

[tex] P(X \geq 7) = [/tex]5.5%

Not good chances!

Part b.)

[tex] P(X \geq 14) = 1- 0.979 [/tex]

[tex] P(X \geq 7) = [/tex]2.1%

Part c.)

[tex] P(X \geq 35) = 1- 0.99870 [/tex]

[tex] P(X \geq 7) = [/tex]0.13%

Part d.)

[tex] P(X \geq 70) = 1- 1[/tex]

I begrudgingly made a quick for loop in MATLAB to calculate Binomial probablity values as high as n=100, with x =70 and p =.5, It spat out 1.000. So,

The probability of getting a 70% and up is 1-1=0. You ant gota chance.

My advice, don't guess on your exams, always cheat.
 
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  • #2
But if you always cheat you will eventually get caught! :)

Here's an auxiliary problem for you. If the probability of getting caught cheating on any given day is 1 in 100, what is the probability that you will be caught cheating over the course of, say, the next 2 years? The next 5 years? The next 10 years?
 
  • #3
NOOO! I seriously have TONS and TONS of stat HW due tomorrow and I am trying to learn as I go because my teacher is terrible :mad: I think I am going to get a 3/7 on my HW if I am LUCKY. Are my answers right?
 
  • #4
I get about 17% for (a) and about 5.8% for (b). I'll give you my other numbers after we figure out why our answers differ.
 

1. How accurate are true/false examinations by tossing a coin?

The accuracy of true/false examinations by tossing a coin depends on the number of questions and the number of tosses. The more questions and tosses, the higher the chance of getting an accurate result.

2. Is it fair to use a coin toss for a true/false examination?

Using a coin toss for a true/false examination can be considered fair if the coin is unbiased and the toss is done in a consistent manner. However, other factors such as the wording of the questions and the subjectivity of the grader can also affect fairness.

3. How does a coin toss compare to other methods of testing true/false knowledge?

A coin toss is a random and unbiased method of testing true/false knowledge. Other methods such as multiple-choice questions or true/false questions with a set number of questions may not be as random or unbiased.

4. Can a coin toss be used for exams in all subjects?

A coin toss can be used for exams in subjects where true/false questions are appropriate. It may not be suitable for subjects that require more complex or in-depth answers.

5. Is there a better alternative to using a coin toss for true/false examinations?

There is no one "best" alternative to using a coin toss for true/false examinations. It ultimately depends on the preferences of the instructor and the nature of the subject being tested. Other alternatives may include multiple-choice questions, true/false questions with a set number of questions, or open-ended questions.

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