IDNLearn.com is your go-to platform for finding reliable answers quickly. Join our community to receive prompt and reliable responses to your questions from experienced professionals.
Sagot :
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]
#### (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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.
