To find the probability of being dealt a card greater than 5 and less than 8 from a standard 52-card deck, we need to carefully consider the range and corresponding cards that meet the criteria.
1. Identify the values of cards that are greater than 5 and less than 8:
- These values are 6 and 7.
2. Determine the count of cards for each value:
- Each value (6 and 7) comes in four suits: spades, hearts, diamonds, and clubs.
- Therefore, there are [tex]\(4\)[/tex] sixes and [tex]\(4\)[/tex] sevens.
3. Calculate the total number of favorable outcomes:
- Number of sixes = 4
- Number of sevens = 4
- Total favorable outcomes = 4 sixes + 4 sevens = 8
4. Determine the total number of possible outcomes:
- The total number of possible outcomes (total cards in the deck) is 52.
5. Calculate the probability:
- Probability [tex]\( P \)[/tex] is given by the ratio of favorable outcomes to total outcomes.
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\][/tex]
- So,
[tex]\[
P = \frac{8}{52}
\][/tex]
6. Simplify the fraction:
- To simplify [tex]\( \frac{8}{52} \)[/tex], we find the greatest common divisor (GCD) of 8 and 52, which is 4.
- Divide the numerator and the denominator by their GCD:
[tex]\[
\frac{8 \div 4}{52 \div 4} = \frac{2}{13}
\][/tex]
Thus, the probability of being dealt a card greater than 5 and less than 8 is [tex]\( \frac{2}{13} \)[/tex].
