Questions: Selecting Cards The red face cards and the black cards numbered 4-9 are put into a bag. Two cards are drawn at random without replacement. Find the following probabilities. Use a TI-83 Plus/TI-84 Plus calculator and round the answers to at least six decimal places. At least 1 of the cards is red.

Solution Steps

To solve this problem, we need to calculate the probability of drawing at least one red card from a specific set of cards. First, identify the total number of red face cards and black cards numbered 4-9. Then, calculate the probability of drawing two cards without replacement such that at least one is red. Use complementary probability to simplify the calculation: find the probability that both cards are black and subtract it from 1.

The red face cards consist of 6 cards (3 from hearts and 3 from diamonds). The black cards numbered 4-9 consist of 12 cards (6 from spades and 6 from clubs). Therefore, the total number of cards is: \[ \text{Total cards} = 6 + 12 = 18 \]

To find the probability that both cards drawn are black, calculate the number of ways to choose 2 black cards from the 12 available, and divide by the number of ways to choose 2 cards from the total of 18: \[ P(\text{both black}) = \frac{\binom{12}{2}}{\binom{18}{2}} = \frac{66}{153} \approx 0.4314 \]

The probability of drawing at least one red card is the complement of the probability of drawing both black cards: \[ P(\text{at least 1 red}) = 1 - P(\text{both black}) = 1 - 0.4314 = 0.5686 \]

Final Answer