Final answer:
To find the probabilities of drawing cards from a bag, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes. The probabilities are as follows: 1 card is red and 1 card is black, all 4 cards are red, 2 cards are red and 2 cards are black, at least one of the red cards is red, all four cards are black.
Step-by-step explanation:
To find the probabilities, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes. Let's go through each probability:
1 card is red and 1 card is black:
There are 3 red face cards and 8 black numbered cards. So, the number of ways to choose 1 red card and 1 black card is 3 * 8 = 24. The total number of cards to choose from is 11 (3 red face cards + 8 black numbered cards). Therefore, the probability is 24 / 11C2.
All 4 cards are red:
There are 3 red face cards and no black numbered cards. So, the number of ways to choose 4 red cards is 3C4 = 0. Therefore, the probability is 0.
2 cards are red and 2 cards are black:
There are 3 red face cards and 8 black numbered cards. So, the number of ways to choose 2 red cards and 2 black cards is 3C2 * 8C2 = 3 * 28 = 84. The total number of cards to choose from is 11 (3 red face cards + 8 black numbered cards). Therefore, the probability is 84 / 11C4.
At least one of the red cards is red:
There are 3 red face cards and 8 black numbered cards. The number of ways to choose at least one red card can be calculated by subtracting the probability of choosing all black cards from 1. The probability of choosing all black cards is (8C2 / 11C2). Therefore, the probability is 1 - (8C2 / 11C2).
All four cards are black:
There are no red face cards and 8 black numbered cards. So, the number of ways to choose 4 black cards is 8C4 = 70. The total number of cards to choose from is 11 (3 red face cards + 8 black numbered cards). Therefore, the probability is 70 / 11C4.