if you get 10 questions correct you get to eat all the green m&m\'s. if you get
ID: 2932750 • Letter: I
Question
if you get 10 questions correct you get to eat all the green m&m's. if you get only 6 correct, you get to eat only the brown peanut M&M's and if you do not get any of the questions correct get to eat 2 m&m's. each question is equally likely to be correct
a) find the probability model for the above scenario
b) find the expected value and standart deviation for the probability model ]
c)find the probability you get at least 5 questions correct
d) find the probability you get at most 7 questions correct
Explanation / Answer
Solution:
a) find probability model for the above scenario
The above scenario can be modelled using Binomial Distribution. The Binomial model is:
P(x; n,p) = nCx px (1-p)n-x x = 0,1,2,....,n
b) find expected value and standard deviation for the probability model
We are given: n = 10, p = 0.5, 1-p = 0.5
We also know that the mean and standard deviation is np and sqrt(np(1-p)) respectively
Therefore Mean = np 10* 0.5 = 5
Standard deviation = sqrt(np(1-p)) = sqrt(10*0.5*0.5) = 1.58
c) find the probability you get at least 5 questions correct
Let X be the number of questions getting correct.
Then we are required to find:
P(X 5) = 1- [P(X = 4)+P(X = 3)+P(X = 2)+ P(X = 1) + P( X= 0)]
= 1- [ 10C4 0.54 (1-0.5)10-4 +10C3 0.53 (1-0.5)10-3 + 10C2 0.52 (1-0.5)10-2+ 10C1 0.51 (1-0.5)10-1 + 10C0 0.50 (1-0.5)10-0]
= 1- [0.2050+ 0.1171+ 0.0439+ 0.0097+ 0.0009]
= 1- [0.3766] = 0.6234
d) find the probability you get at most 7 questions correct
Let X be the number of questions getting correct.
Then we are required to find:
P(X 7) = 1- P(X > 7)
= 1- [P(x = 8) + P(X = 9) +P(X = 10)]
= 1- [ 10C8 0.58 (1-0.5)10-8 +10C9 0.59 (1-0.5)10-9 + 10C10 0.510 (1-0.5)10-10]
= 1- [45* 0.0039*0.25 +10 *0.0019 *0.5 + 1*0.0009*1]
= 1- [ 0.04394+ 0.0097+ 0.0009]
= 1- [0.0544] = 0.945
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