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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.If the die were fair,determine whether or not the events A and B are independent.

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A)
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• If$$\ P(A)-P(B)=P(A\cap$$B)
• Then and B are independent
• Die thrown 2 times
• sample space has 36 events
• S=$${(1,1)(2,2),\dots(6,6)}$$
• A=same number in each throw
• =$${(1,1)(2,2),\dots(6,6)}$$
• B=total of 10 more
• =$${(5,5),(4,6),(6,4),(6,5),(5,6),(6,6)}$$
• $$A\cap$$B=$${(5,5),(6,6)}$$
P(A)=$$p(1,1)+p(2,2)\dots+p(6,6)$$
=$$\frac{1}{6}$$x$$\frac{1}{6}$$+$$\frac{1}{6}$$x$$\frac{1}{6}$$..+$$\frac{1}{6}$$x$$\frac{1}{6}$$
=$$\frac{6}{36}$$
=$$\frac{1}{6}$$
P(B)=$$\frac{6}{36}$$
=$$\frac{1}{6}$$
P(A$$\cap$$B)=$$\frac{2}{36}$$
=$$\frac{1}{12}$$
P(A)P(B)=$$\frac{1}{6}$$x$$\frac{1}{6}$$
=$$\frac{1}{36}$$
$$P(A)P(B)\neq P(A\cap$$B)
AandB are not independent