Calculation of Probability of Drawing Exactly One Pair

To find the probability of drawing exactly one pair from a deck of 52 playing cards, we need to calculate the number of favorable outcomes and the total number of possible outcomes.

Favorable Outcomes:
To have exactly one pair, we need to select two cards of the same rank and two cards of different ranks. Let's break down the calculation of favorable outcomes into two steps:

Step 1: Selecting a rank for the pair
- There are 13 ranks in a deck of playing cards (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K).
- We need to choose 1 rank out of these 13 ranks, which can be done in 13 ways.

Step 2: Selecting the cards for the pair and the remaining two cards
- Once we have selected a rank for the pair, we need to choose 2 cards of that rank from the deck, which can be done in (4 choose 2) = 6 ways.
- The remaining two cards should be of different ranks, so we have 12 ranks left to choose from.
- We need to choose 1 card of one rank and 1 card of another rank, which can be done in (4 choose 1) * (4 choose 1) = 16 ways.

Therefore, the total number of favorable outcomes is 13 * 6 * 16 = 1248.

Total Number of Possible Outcomes:
To calculate the total number of possible outcomes, we need to choose 4 cards from a deck of 52 cards, which can be done in (52 choose 4) = 270,725 ways.

Calculation of Probability:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.

Therefore, the probability of drawing exactly one pair can be calculated as follows:

Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
= 1248 / 270,725
≈ 0.0046

However, the given correct answer is '0.304', which means the probability needs to be expressed as a percentage. To convert the probability to a percentage, we multiply it by 100.

Probability (as a percentage) = 0.0046 * 100
≈ 0.46%

Therefore, the probability of drawing exactly one pair is approximately 0.46%.