Question

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as:
number greater than 4

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Solution

Let X be the random variable, which takes values 0, 1, 2.

therefore space space space straight P left parenthesis straight S right parenthesis space equals space straight P left parenthesis 5 space or space 6 right parenthesis space equals space 2 over 6 space equals space 1 third

straight P left parenthesis straight S with bar on top right parenthesis space equals space 1 minus 1 third space equals space 2 over 3

therefore space space space space space space straight P left parenthesis straight X space equals space 0 right parenthesis space equals space straight P left parenthesis straight S with bar on top. space straight S with bar on top right parenthesis space equals space 2 over 3 cross times 2 over 3 space equals 4 over 9

straight P left parenthesis straight X space equals space 1 right parenthesis space equals space straight P left parenthesis straight S. space straight S with bar on top right parenthesis space plus space straight P left parenthesis straight S with bar on top. space straight S right parenthesis space equals space 1 third cross times 2 over 3 plus 2 over 3 cross times 1 third space equals 4 over 9 straight P left parenthesis straight X space equals space 2 right parenthesis space equals space straight P left parenthesis straight S. space straight S right parenthesis space equals space 1 third cross times 1 third space equals space 1 over 9

∴ X takes the values 0, 1, 2 with probabilities

4 over 9 comma space 4 over 9 space and space 1 over 9

respectively.

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