1a) How many different 4 card hands can be dealt from a deck of 52 cards?   *HINT 11b) How many different 4 card hands can be formed with at least 3 red cards? *HINT 21c) What is the probability of being randomly dealt a 4 card hand with at least 3 red cards? Round to at least 4 decimal places.1d) What is the probability of being randomly dealt a 4 card hand with less than 3 red cards?*HINT 3 Round to at least 4 decimal places.

1a) The number of different 4 card hands that can be dealt from a deck of 52 cards can be calculated using the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items, and r is the number of items to choose. In this case, n = 52 (the total number of cards in the deck) and r = 4 (the number of cards in the hand). So, the number of different 4 card hands is C(52, 4) = 270,725.

1b) To find the number of different 4 card hands that can be formed with at least 3 red cards, we need to consider two cases: exactly 3 red cards and exactly 4 red cards. There are 26 red cards in a deck of 52 cards. So, the number of ways to choose exactly 3 red cards is C(26, 3) = 2,600 and the number of ways to choose the remaining card from the non-red cards is C(26, 1) = 26. Therefore, the number of ways to choose exactly 3 red cards is 2,600 * 26 = 67,600. The number of ways to choose exactly 4 red cards is C(26, 4) = 14,950. So, the total number of 4 card hands with at least 3 red cards is 67,600 + 14,950 = 82,550.

1c) The probability of being randomly dealt a 4 card hand with at least 3 red cards is the number of favorable outcomes (the number of 4 card hands with at least 3 red cards) divided by the total number of outcomes (the total number of 4 card hands). So, the probability is 82,550 / 270,725 = 0.3050 (rounded to 4 decimal places).

1d) The probability of being randomly dealt a 4 card hand with less than 3 red cards is 1 minus the probability of being dealt a hand with at least 3 red cards. So, the probability is 1 - 0.3050 = 0.6950 (rounded to 4 decimal places).