Math Problem Statement

You pick 4 digits (0-9) at random without replacement, and write them in the order picked. What is the probability that you have written the first 4 digits of your phone number? Assume there are no repeats of digits in your phone number.

Solution

This problem is about finding the probability of selecting 4 digits at random without replacement, in such a way that the digits match the first four digits of your phone number, assuming no repeats in your phone number.

Step-by-step solution:

  1. Total digits to choose from: The digits are taken from 0-9, so there are 10 possible digits.

  2. Order matters: Since you're picking 4 digits and writing them in the order picked, the specific arrangement matters. This means we are dealing with permutations, not combinations.

  3. Probability calculation:

    • The first digit of your phone number must match one of the 10 digits you can pick from. So, the chance of picking the correct first digit is 110\frac{1}{10}.
    • After picking the first correct digit, there are only 9 digits left (since we pick without replacement). The chance of picking the correct second digit is 19\frac{1}{9}.
    • For the third digit, there are 8 digits remaining, so the probability of picking the correct third digit is 18\frac{1}{8}.
    • For the fourth digit, there are 7 digits left, so the chance of picking the correct fourth digit is 17\frac{1}{7}.
  4. Multiply these probabilities together: P=110×19×18×17=15040P = \frac{1}{10} \times \frac{1}{9} \times \frac{1}{8} \times \frac{1}{7} = \frac{1}{5040}

So, the probability of selecting the first four digits of your phone number in order is 15040\frac{1}{5040}.

Would you like further details or have any questions?

Related Questions:

  1. What would the probability be if you were picking 3 digits instead of 4?
  2. How would the probability change if repetition of digits were allowed?
  3. What is the probability of selecting any 4 digits in random order without replacement?
  4. How does the number of possible arrangements of 4 digits change if the total digits available increase to 15?
  5. How many different ways can you arrange the 4 digits if they are drawn from 10 digits?

Tip: When solving probability questions with permutations, always consider whether the order matters and whether replacement is involved!

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Math Problem Analysis

Mathematical Concepts

Probability
Permutations

Formulas

P = 1/10 * 1/9 * 1/8 * 1/7

Theorems

Basic Probability Theorem

Suitable Grade Level

Grades 9-12