Does the ticket number contain a seven?

Question

A raffle assigns each entrant a random 4-digit code from 0000 to 9999, with every digit uniform and independent. What is the probability that a code contains the digit 7 at least once?

Accepted Answer

"At least one 7" invites over-counting if you tackle it head-on (a code might have two or three 7s). Use the complement, "no 7 at all." Each digit avoids 7 with probability 	frac{9}{10}, and the four digits are independent, so P(	ext{no 7}) = \left(\frac{9}{10}ight)^4 = \frac{6561}{10{,}000} = 0.6561. Therefore P(	ext{at least one 7}) = 1 - 0.6561 = 0.3439. About 34% of codes show a 7 somewhere. Takeaway. For "at least one," flip to the complement "none," where the independent per-digit failures simply multiply. This dodges the messy inclusion–exclusion over which positions hold the spec…

Does the ticket number contain a seven?

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