# What is the probability that a seven-digit phone number contains the number 7?

Question
Probability
What is the probability that a seven-digit phone number contains the number 7?

2021-01-16
Assuming that the number 7 must appear at least once, we find its complement first, which is not a phone number with number 7. The probability of not having a 7 is 9/10 (there are 10 possible digits). Hence,
$$\displaystyle{P}{\left({n}{o}{7}\right)}={\left(\frac{{9}}{{10}}\right)}\cdot{\left(\frac{{9}}{{10}}\right)}\cdot{\left(\frac{{9}}{{10}}\right)}\cdot{\left(\frac{{9}}{{10}}\right)}\cdot{\left(\frac{{9}}{{10}}\right)}\cdot{\left(\frac{{9}}{{10}}\right)}\cdot{\left(\frac{{9}}{{10}}\right)}={\left(\frac{{9}}{{10}}\right)}^{{7}}$$
Hence, the probability that there is at least 1 number 7 is:
P(at least one 7)=1-P(no 7)
P(at least one 7)=$$\displaystyle{1}-{\left(\frac{{9}}{{10}}\right)}^{{7}}$$
P(at least one 7)$$\displaystyle\sim{0.522}\to{52.2}\%$$

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