Let's solve each part of the given problem step-by-step.
### a. Probability that the card is an Ace
A standard deck of 52 cards includes 4 Aces (one for each suit: hearts, diamonds, clubs, and spades).
The probability that the card selected is an Ace is calculated as:
[tex]\[ P(\text{Ace}) = \frac{\text{Number of Aces}}{\text{Total number of cards}} = \frac{4}{52} \][/tex]
which simplifies to approximately:
[tex]\[ P(\text{Ace}) \approx 0.07692 \][/tex]
So, the probability that the card is an Ace is [tex]\( \approx 0.07692 \)[/tex].
### b. Probability that the card is a face card
Face cards are the Jack, Queen, and King of each suit. There are 3 face cards in each suit and 4 suits in total, so there are:
[tex]\[ 3 \text{ face cards per suit} \times 4 \text{ suits} = 12 \text{ face cards} \][/tex]
The probability that the card selected is a face card is:
[tex]\[ P(\text{Face card}) = \frac{12}{52} \][/tex]
which simplifies to approximately:
[tex]\[ P(\text{Face card}) \approx 0.23077 \][/tex]
So, the probability that the card is a face card is [tex]\( \approx 0.23077 \)[/tex].
### c. Probability that the card is a heart
A standard deck of 52 cards is divided into 4 suits (hearts, diamonds, clubs, and spades), each containing 13 cards.
The probability that the card selected is a heart is:
[tex]\[ P(\text{Heart}) = \frac{13}{52} \][/tex]
which simplifies to:
[tex]\[ P(\text{Heart}) = 0.25 \][/tex]
So, the probability that the card is a heart is [tex]\( 0.25 \)[/tex].
### d. Probability that the card is a black card
Black cards include all the clubs and spades in the deck. Each suit (clubs and spades) contains 13 cards, so there are:
[tex]\[ 13 \text{ cards per suit} \times 2 \text{ suits} = 26 \text{ black cards} \][/tex]
The probability that the card selected is a black card is:
[tex]\[ P(\text{Black card}) = \frac{26}{52} \][/tex]
which simplifies to:
[tex]\[ P(\text{Black card}) = 0.5 \][/tex]
So, the probability that the card is a black card is [tex]\( 0.5 \)[/tex].
### e. Probability that the card shows a number less than 7
Cards showing numbers less than 7 are 2, 3, 4, 5, and 6. Each of these numbers appears once per suit, with 4 suits in total, so there are:
[tex]\[ 5 \text{ cards (less than 7) per suit} \times 4 \text{ suits} = 20 \text{ such cards} \][/tex]
The probability that the card selected shows a number less than 7 is:
[tex]\[ P(\text{Number less than 7}) = \frac{20}{52} \][/tex]
which simplifies to approximately:
[tex]\[ P(\text{Number less than 7}) \approx 0.38462 \][/tex]
So, the probability that the card shows a number less than 7 is [tex]\( \approx 0.38462 \)[/tex].
In summary:
a. [tex]\( P(\text{Ace}) \approx 0.07692 \)[/tex]
b. [tex]\( P(\text{Face card}) \approx 0.23077 \)[/tex]
c. [tex]\( P(\text{Heart}) = 0.25 \)[/tex]
d. [tex]\( P(\text{Black card}) = 0.5 \)[/tex]
e. [tex]\( P(\text{Number less than 7}) \approx 0.38462 \)[/tex]
