Math  /  Data & Statistics

QuestionJulia sets up a passcode on her smart phone, which allows only eight-digit codes. A spy sneaks a look at Julia's smart phone and sees her fingerprints on the screen over eight numbers. What is the probability the spy is able to unlock the smart phone on his first try? Express your answer as a fraction in lowest terms or a decimal rounded to the nearest millionth.

Studdy Solution

STEP 1

What is this asking? If a spy sees eight smudged numbers on a phone, what are the chances they can guess the correct eight-digit passcode on their first try? Watch out! Don't mix up *combinations* and *permutations*!
The order of the digits absolutely matters in a passcode.

STEP 2

1. Find the total number of possible passcodes.
2. Find the number of correct passcodes.
3. Calculate the probability.

STEP 3

We know there are eight possible digits smudged on the screen.
Let's think about how many choices the spy has for each digit in the passcode.

STEP 4

For the first digit, the spy has **8** choices.
For the second digit, they also have **8** choices, and so on for all eight digits in the passcode.
Why? Because they can reuse the smudged numbers!

STEP 5

To find the **total number of possible passcodes**, we multiply the number of choices for each digit together.
This gives us 88888888=888 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 = 8^8.
That's a lot of possibilities!
Calculating this out, we get 88=16,777,2168^8 = \textbf{16,777,216}.

STEP 6

There's only **one** correct passcode!

STEP 7

Probability is calculated as the number of successful outcomes divided by the total number of possible outcomes.

STEP 8

In this case, the number of successful outcomes is **1** (the correct passcode).
The total number of possible outcomes is what we calculated earlier, which is **16,777,216**.

STEP 9

So, the probability is 116,777,216\frac{1}{16,777,216}.
As a decimal, this is approximately **0.00000006**.

STEP 10

The probability the spy unlocks the phone on the first try is 116,777,216\frac{1}{16,777,216}, or approximately 0.00000006.

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