Two cards are drawn in succession from a deck without replacement. What is the probability that both cards are greater than 2 and less than 8?
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To find the probability that both cards are greater than 2 and less than 8, we need to determine the number of favorable outcomes (cards that meet the given criteria) and divide it by the total number of possible outcomes.
First, let's consider the number of favorable outcomes. There are 5 cards greater than 2 and less than 8 in each suit (3, 4, 5, 6, and 7). Since there are 4 suits in a standard deck, the total number of favorable outcomes is 5 cards × 4 suits = 20 cards.
Next, we need to calculate the total number of possible outcomes. When we draw a card without replacement, the total number of possible outcomes decreases with each card drawn. For the first card, we have 52 options (since there are 52 cards in a deck). However, for the second card, we have 51 options since one card has already been drawn.
Therefore, the total number of possible outcomes is 52 × 51.
To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 20 / (52 × 51)
Simplifying the expression, we find:
Probability = 4 / 663
So, the probability that both cards drawn are greater than 2 and less than 8 is 4/663, or approximately 0.00603.