Assume that 5 cards are dealt at random from a standard deck of52 cards (there are 4 suits in the deck and 13 different values(ranks) per each suit). We refer to these 5 cards as a hand in therest of this problem. Calculate the probability of each of thefollowing events when dealing a 5-card hand at random. (a) Exactlyone pair: This occurs when the cards have numeric values a, a, b,c, d, where a, b, c, and d are all distinct. (b) Exactly two pairs:This occurs when the cards have numeric values a, a, b, b, c, wherea, b, and c are all distinct. (c) Only three of a kind: This occurswhen the cards have numeric values a, a, a, b, c, where a, b, and care all distinct. (d) Four of a kind: This occurs when the cardshave numeric values a, a, a, a, b (clearly, b must be differentfrom a because there are only 4 suits in the deck). (e) Full house:This occurs when the cards have numeric values a, a, a, b, b, wherea and b are distinct. (f) Any of the scenarios above will lead tohaving at least a pair in the hand, and having at least a pair inthe hand implies one of the events above must be true. Now, use theprobabilities calculated in parts (a)–(e) to calculate theprobability that we see at least a pair in the hand. Your answerhas to be exactly 49.29%, ignoring rounding error.
