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Question
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
Solution
Let the event be defined as follows:
E1=The die shows 1 or 2
E2=The die shows 3, 4, 5 or 6
E=One of the ball
P(E1)=2/6=1/3 and P(E2)=4/6=2/3 drawn is red and another is black
The probability of drawing a red and a black ball from bag A is given by
`P(E|E_1)=6/10xx4/9+4/10xx6/9=8/15`
The probability of drawing a red and a black ball from bag B is given by
`P(E|E_2)=3/10xx7/9+7/10xx3/9=7/15`
Using the theorem of total probability, we have
`P(E)=P(E_1)P(E|E_1)+P(E_2)P(E|E_2) `
`=1/3xx8/15+2/3xx7/15 `
`=22/45`
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