A standard deck of 52 playing cards contains 13 cards in each of four suits: hearts, diamonds, clubs, and spades. Four cards are drawn from the deck at random. What is the approximate probability that exactly three of the cards are diamonds? 1% 4% 11% 44%
The correct answer and explanation is:
Correct Answer: 4%
To find the approximate probability that exactly three of the four randomly drawn cards are diamonds from a standard 52-card deck, we apply the concept of combinations and probability of compound events.
A standard deck has 13 diamonds and 39 non-diamonds (from the remaining suits).
We are selecting 4 cards such that exactly 3 are diamonds and 1 is not. This is a classic case of the hypergeometric distribution.
The number of ways to choose exactly 3 diamonds from the 13 available is: (133)\binom{13}{3}
The number of ways to choose 1 non-diamond from the remaining 39 cards is: (391)\binom{39}{1}
So, the total favorable outcomes for our event (3 diamonds and 1 non-diamond) is: (133)×(391)\binom{13}{3} \times \binom{39}{1}
Now, we calculate the total number of ways to choose any 4 cards from the full deck: (524)\binom{52}{4}
Now plug in the values:
- (133)=286\binom{13}{3} = 286
- (391)=39\binom{39}{1} = 39
- (524)=270725\binom{52}{4} = 270725
Favorable outcomes: 286×39=11154286 \times 39 = 11154
Total outcomes: 270725270725
Now, compute the probability: P=11154270725≈0.0412 or 4.12%P = \frac{11154}{270725} \approx 0.0412 \text{ or } 4.12\%
This gives an approximate probability of 4%.
This means that when drawing 4 cards at random from a standard deck, there is about a 4% chance that exactly three of them will be diamonds and one will be from any other suit. This relatively low probability reflects the unlikelihood of such a specific outcome in a random draw.