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Question
CBSEENMA12033801

Find the probability distribution of the number of sixes in three tosses of a die.

Solution
Let X denote the number of sixes obtained. X takes the values 0, 1, 2, 3.
Probability of 'getting a six' in a single toss of a die = 1 over 6
therefore space space space space straight p space space space straight i. straight e. comma space space space probability space of space success space equals space 1 over 6
        q i.e., probability of failure  = 1 minus 1 over 6 space equals space 5 over 6
therefore space space space straight P left parenthesis straight X space equals space 0 right parenthesis space equals space straight q space cross times space straight q space cross times space straight q space space equals space 5 over 6 cross times 5 over 6 cross times 5 over 6 space equals space 125 over 216
straight P left parenthesis straight X space equals space 1 right parenthesis space space equals space straight p space cross times space straight q space cross times straight q space plus space straight q space cross times space straight p space space cross times space straight q space plus space straight q space cross times space straight q space cross times space straight p
space space space space space space space space space space space space space space space space space space space equals 1 over 6 cross times 5 over 6 cross times 5 over 6 plus 5 over 6 cross times 1 over 6 cross times 5 over 6 plus 5 over 6 cross times 5 over 6 cross times 1 over 6 equals 25 over 216 plus 25 over 216 plus 25 over 216 equals 75 over 216
straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight p space cross times space straight p space cross times space straight q space plus space straight p space cross times space straight q space cross times straight p space plus space straight q space cross times space straight p space cross times space straight p
space space space space space space space space space space space space space space space space space space space equals space 1 over 6 cross times 1 over 6 cross times 5 over 6 plus 1 over 6 cross times 5 over 6 cross times 1 over 6 plus 5 over 6 cross times 1 over 6 cross times 1 over 6 space equals 5 over 216 plus 5 over 216 plus 5 over 216 equals space 15 over 216
straight P left parenthesis straight X space equals space 3 right parenthesis space equals space straight p space cross times space straight p space cross times space straight p space equals space 1 over 6 cross times 1 over 6 cross times 1 over 6 space equals space 1 over 216
∴ probability distribution is

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