Joe dealt 20 cards from a standard 52-card deck, and the number of red cards exceeded the number of black cards by 8. He reshuffled the cards and dealt 30 cards. This time, the number of red cards exceeded the number of black cards by 10. Determine which deal is closer to the 50/50 ratio of red/black expected of fairly dealt hands from a fair deck and why.
A. The first series is closer because 1/10 is farther from 1/2 than is 1/8.
B. The series closer to the theoretical 50/50 cannot be determined unless the number of red and black cards for each deal is given.
C. The second series is closer because 20/30 is closer to 1/2 than is 14/20.
D. The first series is closer because the difference between red and black is smaller than the difference in the second series.
I NEED HELP PLEASE I THINK THE CORRECT ANSWER IS C
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To determine which deal is closer to the expected 50/50 ratio of red/black cards, we need to compare the proportions of red and black cards in each deal.
In the first deal, Joe dealt 20 cards and found that the number of red cards exceeded the number of black cards by 8. This means he had 14 black cards and 22 red cards out of the 20 dealt. The proportion of red cards to total cards is 22/20 = 1.1, and the proportion of black cards to total cards is 14/20 = 0.7.
In the second deal, Joe reshuffled the cards and dealt 30 cards. This time, the number of red cards exceeded the number of black cards by 10. So, he had 10 black cards and 20 red cards out of the 30 dealt. The proportion of red cards to total cards is 20/30 = 0.67, and the proportion of black cards to total cards is 10/30 = 0.33.
Now, let's analyze the options:
A. The first series is closer because 1/10 is farther from 1/2 than is 1/8.
This option compares the differences between the given ratios (1/10 and 1/8), which is not a suitable method to determine which deal is closer to the expected ratio.
B. The series closer to the theoretical 50/50 cannot be determined unless the number of red and black cards for each deal is given.
This option states that we need more specific information to make a determination. However, we have the necessary information about the numbers of red and black cards in each deal to make a comparison.
C. The second series is closer because 20/30 is closer to 1/2 than is 14/20.
This option correctly compares the proportions of red and black cards in each deal. It states that 20/30 (0.67) is closer to 1/2 than 14/20 (0.7), which implies that the second deal is closer to the expected 50/50 ratio.
D. The first series is closer because the difference between red and black is smaller than the difference in the second series.
This option focuses on the magnitude of the differences between red and black cards in each deal. However, it does not consider the proportions and does not provide a reliable comparison.
Based on the analysis, the correct answer is indeed C. The second series is closer because the proportion of red cards to total cards, 20/30 (0.67), is closer to the expected 50/50 ratio (1/2) than the proportion in the first series (14/20 = 0.7).