Let's solve this step by step:
(a) Compute the probability that the card is an 8.
In a standard deck of 52 cards, there are four 8s (one in each suit: hearts, diamonds, clubs, and spades).
The probability [tex]\(P(\text{8})\)[/tex] can be computed by dividing the number of 8s by the total number of cards in the deck:
[tex]\[ P(\text{8}) = \frac{\text{Number of 8s}}{\text{Total number of cards}} = \frac{4}{52} \][/tex]
Simplifying this fraction:
[tex]\[ P(\text{8}) = \frac{1}{13} \approx 0.07692307692307693 \][/tex]
(b) Compute the probability that the card is a club.
In a standard deck, there are 13 clubs.
The probability [tex]\(P(\text{Club})\)[/tex] can be computed by dividing the number of clubs by the total number of cards in the deck:
[tex]\[ P(\text{Club}) = \frac{\text{Number of clubs}}{\text{Total number of cards}} = \frac{13}{52} \][/tex]
Simplifying this fraction:
[tex]\[ P(\text{Club}) = \frac{1}{4} = 0.25 \][/tex]
(c) Compute the probability that the card is the 8 of clubs.
There is only one 8 of clubs in the entire deck.
The probability [tex]\(P(\text{8 of clubs})\)[/tex] can be computed by dividing the number of 8 of clubs by the total number of cards in the deck:
[tex]\[ P(\text{8 of clubs}) = \frac{\text{Number of 8 of clubs}}{\text{Total number of cards}} = \frac{1}{52} \][/tex]
This simplifies to:
[tex]\[ P(\text{8 of clubs}) \approx 0.019230769230769232 \][/tex]
(d) Compute the probability that the card is not the 8 of clubs.
The probability that a card is not the 8 of clubs can be found by subtracting the probability of getting the 8 of clubs from 1 (since the probabilities of all events must sum to 1):
[tex]\[ P(\text{Not 8 of clubs}) = 1 - P(\text{8 of clubs}) \][/tex]
Using the previously calculated probability for the 8 of clubs:
[tex]\[ P(\text{Not 8 of clubs}) = 1 - \frac{1}{52} \][/tex]
This simplifies to:
[tex]\[ P(\text{Not 8 of clubs}) = \frac{51}{52} \approx 0.9807692307692307 \][/tex]
So, summarizing the results:
- The probability that the card is an 8: [tex]\( \approx 0.07692307692307693 \)[/tex]
- The probability that the card is a club: [tex]\( 0.25 \)[/tex]
- The probability that the card is the 8 of clubs: [tex]\( \approx 0.019230769230769232 \)[/tex]
- The probability that the card is not the 8 of clubs: [tex]\( \approx 0.9807692307692307 \)[/tex]
