In a 10-question true/false test, what is the probability of guessing correctly on questions 1 through 4 exactly 2 times?

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jgardner33v4 · Accepted Answer

Step 1&amp;nbsp;The binomial distribution will apply to a random variable, if it satisfies the conditions given below.&amp;nbsp;1. The experiment can only have two possible results.2. The total number of trials are fixed.&amp;nbsp;3. Each trial is independent.&amp;nbsp;4. Probability of success for each trial is equal.&amp;nbsp;The binomial random variable x is number of successful trials out of total n trials. The probability mass function for binomial distribution is given by&amp;nbsp;P(x)=(nx)px(1−p)n−x, where p is probability of success in each trial.&amp;nbsp;Step 2&amp;nbsp;There are total 4 questions which is total number of trials. Each question has 2 options from which only 1 is correct. Thus, the possible outcomes are correct and incorrect. The probability to choose correct option is&amp;nbsp;12=0.5&amp;nbsp;and each question is independent to other question.&amp;nbsp;Therefore, the number of correct answers is a binomial random variable with&amp;nbsp;n=4&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;p=0.5. To find the probability that 2 question are correct that means number of success is 2, substitute&amp;nbsp;x=2,n=4&amp;nbsp;and&amp;nbsp;p=0.5&amp;nbsp;in&amp;nbsp;P(x)=(nx)px(1−p)n−x.&amp;nbsp;P(2)=(42)(0.5)2(1−0.5)4−2&amp;nbsp;=4!2!(4−2)!(0.25)(0.5)2&amp;nbsp;=4⋅3⋅2!2!⋅2!(0.25)(0.25)&amp;nbsp;=6⋅0.0625&amp;nbsp;=0.375&amp;nbsp;Hence, probability of guessing 2 out of 4 questions is 0.375.

temzej9 · Accepted Answer

Step 1  The score is binomial distributed  (n=10, p=0.5)  The binomial  pdf=(n ch∞se k)pk(1−p)n−k  First, notice that  p=1−p  at p−0.5  Formula simplifies to  (n ch∞se k)pn  Sum that over  7≤k≤10  So,  (10 ch∞se 7)0.510+(10 ch∞se 8)0.510+(10 ch∞se 9)0.510+(10 ch∞se 10)0.510  Simplifying to:  0.510×[(10 ch∞se 7)+(10 ch∞se 8)+(10 ch∞se 9)+(10 ch∞se 10)]  So  10 ch∞se 7=10!×7!×3!  Simplifying  0.510×10!47!×3!×8!×2!×9!×1!×10!×1!  or use your favorite tool, Python, Excel, R. Excel is =1−BINOM.DIST(7,10,0.5,TRUE)=0.0546875

nick1337 · Accepted Answer

To find the probability that something with probability p happens x times in n trials you use this formula: ((n),(x))(p)x(1−p)n−x So we’re looking for the odds that we get 8, 9, or 10 guesses right. ((10),(8))(0.5)8(1−0.5)10−8+((10),(9))(0.5)9(1−0.5)10−9+((10),(10))(0.5)10(1−0.5)10−10 (45)(0.5)10+(10)(0.5)10+(0.5)10 (56)(0.5)10≈0.0635

In a 10-question true/false test, what is the probability of

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