0. [Counting lots of doubles] Roll a pair of dice 102 times. What is the chance

Question

0. [Counting lots of doubles] Roll a pair of dice 102 times. What is the chance of getting doubles at least 20 times? 31. [Groups of scribes Suppose scribes are divided into three groups: 20% in the Plato group, 30% in the Aristotle group, and the rest in the Socrates group. Some of the scribes are left-handed: 5% in the Plato group, 9% in the Aristotle group, and 14% in the Socrates group. If a scribe is selected randomly, what is the chance the scribe is left-handed? If a randomly selected scribe is left-handed, what is the chance the scribe belongs to the Socrates group? a. b.

Explanation / Answer

30)

here probability of getting a double p=6/36 =1/6 (as in 6 events double can happen (1,1),(2,2),..(6,6))

hence expected double in 102 times =np=102*1/6 =17

and std deviaiton =sqrt(np*(1-p))=3.764

hence from normal approximation and continuity correction;

P(getting doubles at least 20 times)=P(X>=20)=1-P(X<=19)=1-P(Z<(19.5-17)/3.764)

=1-P(Z<0.66)=1-0.7454 =0.2546

31)

a)P(left handed)=P(Plato grup and left handed)+P(Aristotle grup and left handed)+P(Socrates grup and left handed)=0.2*0.05+0.3*0.09+(1-0.2-0.3)*0.14=0.107

b)

P(Socrates group|left handed)=P(Socrates grup and left handed)/P(left handed)

=(1-0.2-0.3)*0.14/0.107=0.6542

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