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Bonnie created a website where users could randomly generate a ＂strong password.＂ She claims that $69\%$ of generated passwords contain at least one capital letter, number, and symbol. $\newline$ If the claim is correct, and a user randomly generates $2$ passwords on the website, what is the probability that $0$ of the passwords contain at least one capital letter, number, and symbol? $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____ $\newline$

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Find probabilities using the binomial distribution

Full solution

Q. Bonnie created a website where users could randomly generate a ＂strong password.＂ She claims that

69\%

of generated passwords contain at least one capital letter, number, and symbol.

\newline

If the claim is correct, and a user randomly generates

2

passwords on the website, what is the probability that

0

of the passwords contain at least one capital letter, number, and symbol?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

____

\newline

Find Probability: First, we need to find the probability that a password does not contain at least one capital letter, number, and symbol. Since $69\%$ of passwords do contain these, the probability that a password does not contain them is $100\% - 69\% = 31\%$ .
Convert to Decimal: Now, we convert this percentage to a decimal to make calculations easier. So, $31\%$ as a decimal is $0.31$ .
Calculate Probability: Next, we calculate the probability that both passwords generated do not contain at least one capital letter, number, and symbol. Since the events are independent, we multiply the probability of one password not having these features by itself. So, $0.31 \times 0.31$ .
Multiply Probabilities: Doing the multiplication, we get $0.31 \times 0.31 = 0.0961$ .
Round Result: Finally, we round the result to the nearest thousandth as instructed. The rounded decimal is $0.096$ .

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