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# For a loaded die,the probabilities of outcomes are given as under P(1)=P(2)=0.2,P(3)=P(5)=P(6)=0.1 and P(4)=0.3.The die is thrown two times.Let A and B be the events,'same number each time',and 'a total score is 10 or more',respectively.Determine whether or not A and B are independent. Comment
A)
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• If $$p(A)p(B)=p(A\cap B)$$
• Then Aand Bare independent
• die is thrown 2 times
• sample space={(1,1)(1,2).....(6,6)} 36 such events
• A=same number in each throw
• and (1,1)(2,2)(3,3)(4,4) (5,5)(6,6)
• B=total number is 10 or more
• = (5,5)(6,4)(4,6)(6,5)(5,6)(6,6)
• $$A\cap B$$ ={(5,5)(6,6)}
$$A\cap B$$ ={(5,5)(6,6)}\)
$$=(0.2\times0.2)+(0.2\times0.2)+(0.1)\times(0.1) +(0.4)(0.4)+(0.1)(0.1)+0.1\times0.1$$
$$=0.04+0.04+0.01+0.16+0.01+0.01$$
$$P(A)=0.27$$
$$p(B)=p(5,5)+p(6,4)+.....+p(6,6)$$
$$p(B)=(0.1\times0.1)=0.1\times0.3+0.3\times0.1+0.1\times0.1+0.1\times0.1$$
$$=0.01+0.03+0.03+0.03+0.01+0.01$$
$$=0.09$$
$$p(A\cap\; B)=p(5,5)+p(6,6)$$
$$=0.1\times0.1+0.1\times0.1$$
$$=0.01=0.01$$
$$=0.02$$
$$p(A)\;\times\;p(B)=0.27\times0.09=0.0213$$
$$p(A)p(B)\neq p(A\cap B)$$
so A and B are not indipendent