Given:
Three letters are to be placed into three addressed envelopes.
Each envelope receives exactly one letter.
Letters are inserted at random.
Step 1: Find total number of possible arrangements
Total ways of placing 3 letters into 3 envelopes =
3! = 6
Step 2: Find number of arrangements with no letter in its proper envelope
Such arrangements are called derangements.
Number of derangements of 3 objects =
!3 = 2
(These are: (2,3,1) and (3,1,2))
Step 3: Find number of favourable arrangements
Favourable cases = Total arrangements − Derangements
= 6 − 2
= 4
Step 4: Calculate probability
Probability =
(Number of favourable outcomes) / (Total number of outcomes)
= 4 / 6
= 2 / 3
Final Answer:
The probability that at least one letter is in its proper envelope is
2 / 3
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?