What is the probability that the deal of a five-card hand provides exactly one ace?
Step-1: Find the total number of possible hands-in tools:
A standard deck of cards contains cards, of which are red and are black, of each are hearts, diamonds, spades, and clubs.
The order of the cards does not matter (since a different order results in the same hand) and thus we need to use the definition of a combination.
A standard deck of cards contains cards. A five-card poker hand selects of the cards.
Use the combination formula:
.
Where represents the total number of items and represents the number of items being chosen at a time.
Substitute and in the above formula:
So, the total number of possible outcomes are .
Step-2: Find the number of hands containing an ace:
The standard deck of cards contains four aces, so set :
As well as the five cards in our hands, we should consider the other four.
There are cards to pick from, so we can set .
Multiplying the above two equations together, we can get exactly one Ace:
So, the number of successful outcomes .
Step-3: Find the probability that the deal of a five-card hand provides exactly one ace:
The probability is the number of favorable outcomes divided by the number of possible outcomes.
Substitute the values in the above formula:
Hence, the probability that the deal of a five-card hand provides exactly one ace is .