Certainly! Let's break down the problem step-by-step to determine the probabilities for each scenario.
#### (a) [tex]\( P(\text{Queen and a 3}) \)[/tex]
The event of drawing a Queen and a 3 simultaneously is impossible because you can only draw one card at a time from a deck. Thus, the probability of drawing a Queen and a 3 at the same time is:
[tex]\[ P(\text{Queen and a 3}) = 0 \][/tex]
#### (b) [tex]\( P(\text{face card or a number card}) \)[/tex]
A standard deck of 52 cards consists entirely of face cards and number cards. Since every card in the deck is either a face card or a number card, the probability of drawing either a face card or a number card is:
[tex]\[ P(\text{face card or a number card}) = 1 \][/tex]
#### (c) [tex]\( P(\text{black and a Queen}) \)[/tex]
There are 4 Queens in a deck, one in each suit (hearts, diamonds, clubs, and spades). Out of these, two Queens are black (the Queen of clubs and the Queen of spades). The probability of drawing a black Queen is:
[tex]\[ P(\text{black and a Queen}) = \frac{2}{52} = \frac{1}{26} \approx 0.038 \][/tex]
#### (d) [tex]\( P(\text{black or a face card}) \)[/tex]
There are 26 black cards in the deck (13 spades and 13 clubs). There are 16 face cards in the deck (4 Jacks, 4 Queens, 4 Kings, and 4 Aces). The total black face cards are 6 (3 face cards each in spades and clubs). The probability of drawing a black card or a face card involves adding the probabilities of each and subtracting the overlap:
[tex]\[ P(\text{black or a face card}) = \frac{26 + 16 - 6}{52} = \frac{36}{52} = \frac{18}{26} \approx 0.692 \][/tex]
#### (e) [tex]\( P(\text{black and a face card}) \)[/tex]
There are 3 face cards (Jack, Queen, King) for each suit, and since there are 2 black suits (spades and clubs), there are a total of 6 black face cards. Therefore, the probability of drawing a black face card is:
[tex]\[ P(\text{black and a face card}) = \frac{6}{52} = \frac{3}{26} \approx 0.115 \][/tex]
So the answers are:
(a) [tex]\( P(\text{Queen and a 3}) = 0 \)[/tex]
(b) [tex]\( P(\text{face card or a number card}) = 1 \)[/tex]
(c) [tex]\( P(\text{black and a Queen}) \approx 0.038 \)[/tex]
(d) [tex]\( P(\text{black or a face card}) \approx 0.692 \)[/tex]
(e) [tex]\( P(\text{black and a face card}) \approx 0.115 \)[/tex]
