Calculating Probability for 4-Digit Numbers

The problem asks for the probability that a randomly selected 4-digit number, formed using the digits {0, 1, 2, 3, 4} without repetition, does not contain the digit zero.

Total Possible Outcomes: Total 4-Digit Numbers

First, determine the total number of valid 4-digit numbers possible using the digits {0, 1, 2, 3, 4} without repetition.

• The first digit (thousands place) cannot be 0. Thus, there are 4 choices (1, 2, 3, or 4).
• After selecting the first digit, 4 digits remain (including 0).
• The second digit can be any of the remaining 4 digits.
• The third digit can be any of the remaining 3 digits.
• The fourth digit can be any of the remaining 2 digits.
• The total number of 4-digit numbers is the product of the choices for each position: \(4 \times 4 \times 3 \times 2 = 96\).

Alternatively, using permutations:

• Total ways to arrange 4 digits out of 5 is P(5, 4) = \(\frac{5!}{(5-4)!} = 120\).
• Numbers with a leading 0 are not 4-digit numbers. If 0 is the first digit, the remaining 3 digits can be arranged in P(4, 3) = \(\frac{4!}{(4-3)!} = 24\) ways.
• Total valid 4-digit numbers = Total arrangements - Arrangements starting with 0 = 120 - 24 = 96.

Favorable Outcomes: 4-Digit Numbers Without Zero

Next, determine the number of 4-digit numbers formed using only the digits {1, 2, 3, 4} without repetition (i.e., numbers that do not contain zero).

• We must use the 4 digits {1, 2, 3, 4}.
• The number of ways to arrange these 4 distinct digits is P(4, 4) = 4! = \(4 \times 3 \times 2 \times 1 = 24\).

Probability Calculation

The probability is the ratio of the number of favorable outcomes (4-digit numbers without zero) to the total number of possible outcomes (total 4-digit numbers).

Probability = \(\frac{\text{Number of 4-digit numbers without zero}}{\text{Total number of 4-digit numbers}}\)

Probability = \(\frac{24}{96}\)

Simplifying the fraction:

Probability = \(\frac{1}{4}\)

Converting to decimal form:

Probability = 0.25.