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Question
An experiment consists of rolling a die until a 2 appears. How many elements of the sample space correspond to the event that the 2 appears not later than the k th roll of the die?
Solution
Number of sample space = 6
In this case, 2 appears not later than kth roll of the die
Then it is possible that 2 comes in first throw i.e. 1 outcome.
If 2 does not appear in first throw
Then outcomes will be 5 and 2 outcomes in second throw i.e. 1 outcome.
∴ Possible outcome = 5 × 1 = 5
Similarly, if 2 does not appear in second throw and appears in third throw
∴ Possible outcome = 5 × 5 × 1
Now we have the series:
= `1 + 5 + 5 xx 5 + 5 xx 5 xx 5 + ... + 5^(k - 1)`
= `1 + 5 + 5^2 + 5^3 + ... + 5^(k - 1)`
= `(.*(r^k - 1))/(r - 1)`
= `(5^k - 1)/(5 - 1)`
= `(5^k - 1)/4`
Hence, the required answer = `(5^k - 1)/4`.
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