Question 982731
 
Given:
A 4-sided die, with the following pmf:
1: 0.279
2: 0.235
3: 0.266
4: 0.220
(Total=1.000)
 
(a) Rolled 26 times, need probability that an even number occurs exactly 12 times.
(b) Find expected value of a single roll of the die.
 
Solution:
(A) Probability of getting an even number 12 times out of 26 tosses.
Probability of getting an even number at each toss, 
p = P(2)+P(4)=0.235+0.220=0.455
 
To find probability of rolling an even number 12 times out of 26, we use binomial theorem:
{{{P(n,r)=C(n,r)p^r* (1-p)^(n-r)}}}
{{{    where C(n,r)=n!/(r!(n-r)!)}}}
 
Therefore
{{{P(26,12)=C(26,12)*0.455^12* (1-0.455)^(26-12)=0.15508}}}
 
 
(B) Expected value of a single throw E[X]
Using the standard formula,
E[X]=sum x*p(x)={{{1*0.279+2*0.235+3*0.266+4*0.220=2.427}}}