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.
Similar Questions
- What is the probability that getting a total of the numbers on the dice is 6 ?
- What is the probability that getting a total of the numbers on the dice is at least 23 ?
- In a sample survey of a village, the probability that a farmer is in debt is 0.60. What is the probability that three randomly selected farmers are all in debt (assume independence of events)?
- The probability that a family owns a laptop is 0.68; that it also owns a desktop is 0.56. If the probability that it owns both is 0.48, then what is the probability that a randomly selected family owns a laptop or a desktop?
Important Questions from Probability
Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?
If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?
A. 2/3
B. 3/4
C. 1/4
D. 1/9Statements followed by some conclusions are given below.
Statements:
1. A bag has 2 white, 3 black, 4 red and 6 green balls.
2. 1 ball selected at random from the bag.
Conclusions:
I. The probability that a black ball is selected is 1/5
II. The probability that a red ball is selected is 6/15
Find which of the conclusions logically follows from the given statement
A. Only conclusion I follows.
B. Only conclusion II follows.
C. Both I and II follow.
D. Neither I nor II follows.
In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?
A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
