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प्रश्न
Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.
उत्तर
Success is defined as a number greater than 4 appears on at least one die.
Let X denote the number of successes.
∴ Possible values of X and 0, 1, 2.
Let, P(getting a number greater than 4) = p = `(2)/(6) = (1)/(3)`
∴ q = 1 – p = `1 - (1)/(3) = (2)/(3)`
∴ P(X = 0) = No number greater than 4 in two tosses of dice.
∴ P(X = 0) = P {(1, 1) (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}
`= 16/36 = 4/9`
∴ P(X = 1) = Only 1 number is greater than 4 in two tosses of o die.
∴ P(X = 1) = P(one success) = P {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4)}
`= 16/36`
= `(4)/(9)`
∴ P(X = 2) = Two numbers are greater than 4 in two tosses of die.
∴ P(X = 2) = P {(5, 5), (5, 6), (6, 5), (6, 6)}
`= 4/36`
= `1/9`
∴ Probability distribution of X is as follows:
| X | 0 | 1 | 2 |
| P(X = x) | `(4)/(9)` | `(4)/(9)` | `(1)/(9)` |
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