If a gambler rolls 7 dice what is the expected number of 5s

If a gambler rolls 7 dice, what is the expected number of 5\'s and 6\'s? Suppose X is a random variable. How are the variance and standard deviation of X related? The variance of X is the square root of the standard deviation of X

Solution

7 DICE ARE THROWN SO

PROBABILITY OF GETTING 5 AND 6 IN EACH DICE = 1/3

PROBABILITY OF NOT GETTING 5 AND 6 ARE = 2/3

GETTING 0 TIMES 5 AND 6 = 7C0*(1/3)^0*(2/3)^7 = 0.05 EXPECTATION = 0*0.05 = 0

GETTIMG 1 TIME = 7C1*(1/3)^1*(2/3)^6 = 0.40 1*0.40 = 0.40

2 TIMES = 7C2*(1/3)^2*(2/3)^5 = 0.30 2*0.30 = 0.60

3 TIMES = 7C3*(1/3)^3*(2/3)^4 = 0.24 3*0.24 = 0.72

4 TIMES = 7C4*(1/3)^4*(2/3)^3 = 0.11 4*0.11 = 0.44

5 TIMES = 7C5*(1/3)^5*(2/3)^2= 0.03 5*0.03 = 0.15

6 TIMES = 7C6*(1/3)^6*(2/3)^1 = 0.005 6*0.005 = 0.30

7 TIMES = 7C7*(1/3)^7*(2/3)^0 = 0.0004 7*0.0004 = 0.0028

ADD ALL EXPECTATION = 2.6138

OPTION B IS CORRECT