A 4-digit number is selected at random formed by using the digits 0, 1, 2, 3 and 4 (where repetition of digits is not allowed).
We need to find the probability that a randomly selected 4-digit number, formed using the digits 0, 1, 2, 3, 4 without repetition, is divisible by 4.
Calculating Total Possible Numbers
A 4-digit number cannot start with 0. The digits are {0, 1, 2, 3, 4} without repetition.
- First digit: 4 choices (cannot be 0, so 1, 2, 3, 4).
- Second digit: 4 choices (remaining digits including 0).
- Third digit: 3 choices.
- Fourth digit: 2 choices.
The total number of possible 4-digit numbers is calculated as: \( 4 \times 4 \times 3 \times 2 = 96 \) There are 96 possible 4-digit numbers.
Calculating Favorable Outcomes (Divisible by 4)
A number is divisible by 4 if its last two digits form a number divisible by 4. We list the possible pairs of last two digits (XY) using {0, 1, 2, 3, 4} without repetition:
- Ending in 04: Digits used {0, 4}. Remaining {1, 2, 3}. First digit has 3 choices, second has 2. Ways = \(3 \times 2 = 6\).
- Ending in 12: Digits used {1, 2}. Remaining {0, 3, 4}. First digit has 2 choices (not 0). Second digit has 2 choices. Ways = \(2 \times 2 = 4\).
- Ending in 20: Digits used {2, 0}. Remaining {1, 3, 4}. First digit has 3 choices. Second digit has 2. Ways = \(3 \times 2 = 6\).
- Ending in 24: Digits used {2, 4}. Remaining {0, 1, 3}. First digit has 2 choices (not 0). Second digit has 2 choices. Ways = \(2 \times 2 = 4\).
- Ending in 32: Digits used {3, 2}. Remaining {0, 1, 4}. First digit has 2 choices (not 0). Second digit has 2 choices. Ways = \(2 \times 2 = 4\).
- Ending in 40: Digits used {4, 0}. Remaining {1, 2, 3}. First digit has 3 choices. Second digit has 2. Ways = \(3 \times 2 = 6\).
The total number of favorable outcomes (numbers divisible by 4) is the sum of ways for each ending:
6 + 4 + 6 + 4 + 4 + 6 = 30 There are 30 numbers divisible by 4.Calculating the Probability
The probability is the ratio of favorable outcomes to total possible outcomes:
\( P(\text{Number is divisible by 4}) = \frac{\text{Total numbers divisible by 4}}{\text{Total possible 4-digit numbers}} \) \( P = \frac{30}{96} \)Simplifying the fraction:
\( P = \frac{30 \div 6}{96 \div 6} = \frac{5}{16} \)The probability is 5/16.
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Important Questions from Probability
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
The probability of having 53 Tuesdays in an ordinary year is:
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If two balanced dice are tossed once, the probability of the event, that the sum of the integers coming on the upper sides of the two dice is 9, is
