Question

you are dealt one card from a standard 52 - card deck. find the probability of being dealt a card greater than 5 and less than 8. click the icon to view a description of a standard deck of playing cards. the probability of being dealt a card greater than 5 and less than 8 is \\(\square\\). (type an integer or a simplified fraction.)

Explanation:

Step1: Identify valid cards

Cards greater than 5 and less than 8 are 6 and 7. Each has 4 suits (hearts, diamonds, clubs, spades), so total valid cards: \(4 + 4 = 8\)? Wait, no: 6 has 4, 7 has 4, so total \(4\times2 = 8\)? Wait, no, 6, 7: each rank has 4 cards. So number of favorable outcomes \(n = 4\times2 = 8\)? Wait, no, 6 and 7: two ranks, each with 4 suits. So \(n = 2\times4 = 8\)? Wait, no, wait: 5 < card < 8, so card values are 6, 7. Each value (rank) has 4 cards (one for each suit: hearts, diamonds, clubs, spades). So number of favorable cards: \(2\times4 = 8\)? Wait, no, wait: 6, 7: two ranks, each with 4 cards. So 4 + 4 = 8? Wait, no, 6 has 4, 7 has 4, so total 8? Wait, no, wait, standard deck: ranks are 2 - 10, J, Q, K, A. So 6, 7: each has 4 suits. So number of favorable outcomes: 4 (for 6) + 4 (for 7) = 8? Wait, no, wait, 6 and 7: two ranks, each with 4 cards. So 2*4=8. Total number of cards in deck: 52.

Step2: Calculate probability

Probability \(P = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}} = \frac{8}{52}\)? Wait, no, wait: 6, 7: each rank has 4 cards, so 2 ranks 4 cards = 8? Wait, no, wait, 6, 7: that's two ranks, each with 4 suits. So 4 + 4 = 8? Wait, no, 6 has 4 (hearts, diamonds, clubs, spades), 7 has 4. So total 8. Wait, but wait, 5 < card < 8: so card values are 6, 7. So number of favorable cards: 4 (for 6) + 4 (for 7) = 8? Wait, no, 6, 7: each has 4, so 24=8. Total cards: 52. Then simplify \(\frac{8}{52}\) by dividing numerator and denominator by 4: \(\frac{8\div4}{52\div4} = \frac{2}{13}\)? Wait, no, 8 divided by 4 is 2, 52 divided by 4 is 13. Wait, but wait, did I count the favorable cards correctly? Wait, 6, 7: two ranks, each with 4 cards. So 2*4=8. Total cards 52. So probability is \(\frac{8}{52} = \frac{2}{13}\)? Wait, no, wait: 6, 7: each rank has 4 cards, so 4 + 4 = 8. So 8/52 = 2/13. Wait, but let's check again: 5 < card < 8: so card values are 6, 7. Each value has 4 suits. So number of favorable outcomes: 4 (6) + 4 (7) = 8. Total outcomes: 52. So probability is 8/52 = 2/13.

Wait, no, wait, 6, 7: two ranks, each with 4 cards. So 2*4=8. So 8/52 = 2/13. Yes.

Answer:

\(\frac{2}{13}\)