Not sure what i did wrong binomial probability

In summary, the student has a 0.1004 chance of passing based on the normal approximation, but they would be more comfortable if the calculation was exact.
  • #1
mtingt
13
0
In a 22-item true–false examination, a student guesses on each question.

If 14 correct answers constitute a passing grade, what is the probability the student will pass?

i did c(22,14)* (1/2)^14 * (1/2)^8
 
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  • #2
You have to add in the probabilities for more than 14 correct guesses. Your answer is for exactly 14 correct.
 
  • #3
so i would have to add every single probability up to 22?
is there any other way i could do this?
 
  • #4
mtingt said:
so i would have to add every single probability up to 22?
is there any other way i could do this?

You can use an online calculator for p=0.5, n=22, x=14 and solve for [itex]P(X\geq x)[/itex]

http://stattrek.com/Tables/Binomial.aspx
 
  • #5
mtingt said:
so i would have to add every single probability up to 22?
is there any other way i could do this?
You might also use a Normal approximation to the Binomial distribution.
 
  • #6
awkward said:
You might also use a Normal approximation to the Binomial distribution.

The normal approximation gives p=0.1004 whereas the presumably exact binomial gives [itex](P(X\geq x)=0.1431 [/itex] for x=14.

For the normal approximation I'm using mean 11 and [itex] SD = \sqrt {11(1-0.5)} = 2.345 [/itex]
 
  • #7
SW VandeCarr said:
The normal approximation gives p=0.1004 whereas the presumably exact binomial gives [itex](P(X\geq x)=0.1431 [/itex] for x=14.

For the normal approximation I'm using mean 11 and [itex] SD = \sqrt {11(1-0.5)} = 2.345 [/itex]
I get [itex]P(X \geq 13.5) = 0.1432[/itex] using the Normal distribution adjusted for continuity.
 
  • #8
awkward said:
I get [itex]P(X \geq 13.5) = 0.1432[/itex] using the Normal distribution adjusted for continuity.

I did too, but when Ted Williams was told his 0.3995 batting average would go into the record books as 0.400, he said that wasn't really 0.400 and played through two final season games ending up with a 0.406 batting average. Is 13.5 a passing grade or is 14 a threshold value? I understand the continuity correction and it's fine for some applications but for n=22 and a "threshold" value, why not use an exact calculation? In either case, you will likely use tables or a calculator.

Having said that, it's closer than I would have thought, but I wouldn't have been comfortable without doing the exact approach.
 
Last edited:
  • #9
I agree, in this case the approximation works better than we have any right to expect. Still, it's a useful tool to have around.
 

What is binomial probability?

Binomial probability is a mathematical concept that calculates the likelihood of a specific event occurring a certain number of times in a fixed number of trials. It is often used in statistics and probability to analyze experimental results and make predictions.

How is binomial probability calculated?

The formula for binomial probability is P(x) = nCx * p^x * (1-p)^(n-x), where n is the number of trials, x is the number of successes, and p is the probability of success in each trial. nCx represents the number of ways that x successes can occur in n trials.

What is the difference between binomial probability and normal distribution?

Binomial probability is used to calculate the probability of a specific event occurring a certain number of times in a fixed number of trials, while normal distribution is used to analyze continuous data and calculate the probability of a range of values occurring. Additionally, binomial probability assumes a fixed probability of success in each trial, while normal distribution assumes a continuous probability distribution.

What are the conditions for using binomial probability?

There are three conditions that must be met to use binomial probability: 1) the trials must be independent, meaning the outcome of one trial does not affect the outcome of another; 2) there must be a fixed number of trials; and 3) the probability of success must be the same for each trial.

How is binomial probability used in real life?

Binomial probability is used in a variety of real-life situations, such as predicting the outcomes of coin tosses, analyzing the effectiveness of a new medication, and predicting the success of a marketing campaign. It is also used in genetics to analyze the probability of inheriting certain traits.

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