I worked out the problem, i want to check to see if my work is correct.. please
ID: 3206790 • Letter: I
Question
I worked out the problem, i want to check to see if my work is correct.. please show step by step, thx!
1) a exam has 11 multiple choice questions. Each with 5 answer choices. if a studnet takes the exam and randomly quesses on every question.
a) find the probability the studnet gets between 3 and 8, exclusive, questions correct?
b) find the probability the student gets more than 7 questions correct?
2) The lottery has 900 prizes of $1. 140 prizes of $10. 30 prizes of $65, 5 prizes of $290, 2 prizes of $1170, and 1 prize of $2700. Assume that 30,000 lottery tickets are issued and sold for $1.
a) what is the lotterys expected profit per ticket?
b) what is the lottery;s standard deviation of profit per ticket?
Explanation / Answer
Q1.
Binomial Distribution
PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
a.
P( X = 7 ) = ( 11 7 ) * ( 0.2^7) * ( 1 - 0.2 )^4
= 0.0017
P( X = 6 ) = ( 11 6 ) * ( 0.2^6) * ( 1 - 0.2 )^5
= 0.0097
P( X = 5 ) = ( 11 5 ) * ( 0.2^5) * ( 1 - 0.2 )^6
= 0.0388
P( X = 4 ) = ( 11 4 ) * ( 0.2^4) * ( 1 - 0.2 )^7
= 0.1107
P(3 < X < 8) = P(X=7) + P(X=6) + P(X=5) + P(X=4)
= ( 11 7 ) * 0.2^7 * ( 1- 0.2 ) ^4 + ( 11 6 ) * 0.2^6 * ( 1- 0.2 ) ^5 + ( 11 5 ) * 0.2^5 * ( 1- 0.2 ) ^6 + ( 11 4 ) * 0.2^4 * ( 1- 0.2 ) ^7
= 0.0017 + 0.0097 + 0.0388 + 0.1107
= 0.1609
b.
P( X < = 7) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 11 7 ) * 0.2^7 * ( 1- 0.2 ) ^4 + ( 11 6 ) * 0.2^6 * ( 1- 0.2 ) ^5 + ( 11 5 ) * 0.2^5 * ( 1- 0.2 ) ^6 + ( 11 4 ) * 0.2^4 * ( 1- 0.2 ) ^7 + ( 11 3 ) * 0.2^3 * ( 1- 0.2 ) ^8 + ( 11 2 ) * 0.2^2 * ( 1- 0.2 ) ^9 + ( 11 1 ) * 0.2^1 * ( 1- 0.2 ) ^10 + ( 11 0 ) * 0.2^0 * ( 1- 0.2 ) ^11
= 0.9998
P( X > 7) = 1 - P ( X <=7) = 1 -0.9998 = 0.0002