Given:
A standard deck has 52 cards:
Diamonds = 13 cards
Spades = 13 cards
Four cards are drawn at random.
Step 1: Find the total number of ways to draw 4 cards
Total ways =
52C4
Step 2: Find the number of favourable outcomes
To obtain exactly 3 diamonds and 1 spade:
Ways to choose 3 diamonds from 13 =
13C3
Ways to choose 1 spade from 13 =
13C1
Total favourable outcomes =
13C3 × 13C1
Step 3: Calculate the probability
Probability =
( Number of favourable outcomes ) / ( Total number of outcomes )
= ( 13C3 × 13C1 ) / 52C4
Step 4: Simplify (optional)
13C3 = 286
13C1 = 13
52C4 = 270725
Probability =
(286 × 13) / 270725
= 3718 / 270725
Final Answer:
The probability of obtaining 3 diamonds and 1 spade is
( 13C3 × 13C1 ) / 52C4
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?