Given:
Digits available: 0, 1, 3, 5, 7
Four–digit numbers greater than 5000 are formed.
A number is divisible by 5 if its last digit is 0 or 5.
Total possible cases
Since the number must be greater than 5000, the thousand’s digit can be:
5 or 7
(i) Digits may be repeated
Total number of outcomes:
Thousand’s digit → 2 choices (5, 7)
Hundreds digit → 5 choices
Tens digit → 5 choices
Units digit → 5 choices
Total outcomes = 2 × 5 × 5 × 5 = 250
Favourable outcomes (divisible by 5):
Units digit must be 0 or 5 → 2 choices
Thousand’s digit → 2 choices
Hundreds digit → 5 choices
Tens digit → 5 choices
Favourable outcomes = 2 × 5 × 5 × 2 = 100
Probability:
P = 100 / 250 = 2 / 5
(ii) Repetition of digits is not allowed
Total number of outcomes:
Thousand’s digit → 2 choices (5, 7)
Remaining digits for hundreds, tens, units → 4, 3, 2 choices respectively
Total outcomes = 2 × 4 × 3 × 2 = 48
Favourable outcomes (divisible by 5):
Case 1: Units digit = 0
Thousand’s digit → 2 choices
Remaining digits → choose 2 from remaining 3 digits in order
Ways = 2 × 3 × 2 = 12
Case 2: Units digit = 5
Then thousand’s digit must be 7 (since repetition is not allowed)
Remaining digits → choose 2 from remaining 3 digits in order
Ways = 1 × 3 × 2 = 6
Total favourable outcomes = 12 + 6 = 18
Probability:
P = 18 / 48 = 3 / 8
Final Answers:
(i) Probability = 2 / 5
(ii) Probability = 3 / 8
A die is thrown. Describe the following events:
(i) \(A: a\) number less than \(7\)
(ii) \(B: a\) number greater than \(7\)
(iii) \(C: a\) multiple of \(3\)
(iv) \(D: a\) number less than \(4\)
(v) \(E: a\) even number greater than \(4\)
(vi) \(F: a\) number not less than \(\)\(3\)
Also, find \(A∪B, A∩B, B∪C, E∩F, D∩E, A-C, D-E, E∩F', F'\)
An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events: \(A:\) the sum is greater than \(8\), \(B:\)\(2\) occurs on either die \(C:\)The sum is at least \(7\), and a multiple of \(3\). Which pairs of these events are mutually exclusive?