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For a loaded die, the probabilities of o...

For a loaded die, the probabilities of outcomes are given as under: `P(1)=P(2)=2/(10),P(3)=P(5)=P(6)=1/(10)a n dP(4)=3/(10)` The die is thrown two times. Let A and B be the events as defined below A=Getting same number each time, B=Getting a total score of 10 or more. Determine whether or not A and B are independent events.

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for a loaded die, it is given that
P(1)=P(2)=0.2,
P(3)=P(5)=P(6)=0.1 and P(4)=0.3
Also, die is thrown two times.
Here, A = same number each time and B= Total sore is 10 or more
`therefore` A={(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
So, P(A)=[P(1,1)+P(2,2)+P(3,3)+P(4,4)+P(5,5)+P(6,6)]
`=[P(1)cdotP(1)+P(2)cdotP(2)+P(3)cdotP(3)+P(4)cdotP(4)+P(5)cdotP(5)+P(6)cdotP(6)]`
`=[0.2xx0.2+0.2xx0.2+0.1xx0.1+0.3xx0.3+0.1xx0.1+0.1xx0.1]`
=0.04+0.04+0.01+0.09+0.01+0.01=0.20
and B={(4,6),(6,4),(5,5),(5,6),(6,5),(6,6)}
`thereforeP(B)=P(4,6)+P(6,4)+P(5,5)+P(5,6)+P(6,5)+P(6,6)`
`=P(4)cdotP(6)+P(6)cdotP(4)+P(5)cdotP(5)+P(5)cdotP(6)+P(6)cdotP(5)+P(6)cdotP(6)`
`=0.3xx0.1+0.1xx0.2+0.1xx0.1+0.1xx0.1+0.1xx0.1+0.1xx0.1`
`=0.03+0.03+0.01+0.01+0.01+0.01=0.10`
Also,`AcapB={(5,5),(6,6)}`
`thereforeP(AcapB)=P(5,5)+P(6,6)=P(5)cdotP(5)+P(6)cdotP(6)`
`=0.1xx0.1+0.1xx0.1=0.01+0.001=0.02`
We know tht, for two events A and B, if `P(AcapB)=P(A)cdotP(B)`, then both are independent events.
Here, `P(AcapB)=0.02 and P(A)cdotP(B)=0.02`
Hence, A and B are independent events.
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