Question

A die is tossed thrice. a success is ‘getting 1 or 6'on a toss. find the mean and the variance of the number of successes.

Answer 1

Solution :-

Let x be the number of success.

Obviously, x can take the values 0, 1, 2 or 3

Probability of success = 2/6 = 1/3

Probability of failure = 1/1 - 1/3 = 2/3

P(x = 0) = P (no success) = P (all failures) = 2/3*2/3*2/3 = 8/27

P(x = 1) = P (1 success and 2 failures) = 3c1 × 1/3*2/3*2/3 = 12/27

P(x = 2) = P (2 success and 1 failure) = 3c2 × 1/3*1/3*2/3 = 6/27

P(x = 3) = P (all 3 success) = 1/3*1/3*1/3 = 1/27

∴ The probability distribution of the random variable x is -

x       :     0               1           2           3

P(x)  :   8/27        12/27      6/27      1/27

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x₁                             p₁                            p₁x₁                         p₁x₁²
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0                            8/27                             0                              0 

1                           12/27                         12/27                       12/27

2                            6/27                          12/27                       24/27    

3                            1/27                           3/27                         9/27
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                                                                  1                      45/27 = 5/3

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Mean μ = Σp₁x₁ = 1

Variance = б² = Σp₁x₁² - μ

⇒ 5/3 - 1/1 

= 2/3

Answer.