To find the probability of picking a 7 and then an 8 from a standard deck of cards, with replacement, you perform the following steps:
1. Understand the Deck: A standard deck of cards has 52 cards, consisting of 4 suits (hearts, diamonds, clubs, spades). Each suit has 13 ranks (Ace through King). Therefore, there are 4 cards of each rank.
2. Probability of Picking a 7: Since there are 4 sevens in the deck, the probability of picking a 7 is the number of 7s divided by the total number of cards.
[tex]\[ P(\text{7}) = \frac{4}{52} \][/tex]
3. Replacing the Card: After you pick the first card, you put it back in the deck. This means the deck returns to its original count of 52 cards.
4. Probability of Picking an 8: Similarly, since the deck has been replenished, there are still 4 eights in the deck. Thus, the probability of picking an 8 after replacing the first card is also the number of 8s divided by the total number of cards.
[tex]\[ P(\text{8}) = \frac{4}{52} \][/tex]
5. Combined Probability: Since you are picking a 7 and then an 8 with replacement, the two events are independent. The combined probability of two independent events is the product of their individual probabilities.
[tex]\[ P(\text{7 and then 8}) = P(\text{7}) \times P(\text{8}) \][/tex]
Now, let's calculate these probabilities:
[tex]\[ P(\text{7}) = \frac{4}{52} = \frac{1}{13} \][/tex]
[tex]\[ P(\text{8}) = \frac{4}{52} = \frac{1}{13} \][/tex]
The combined probability is:
[tex]\[ P(\text{7 and then 8}) = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169} \][/tex]
So, the probability of picking a 7 and then picking an 8, with replacement, is [tex]\(\frac{1}{169}\)[/tex].
