Karen, Gavin, Nasra, and Ali each have a deck of playing cards. Each student randomly draws a card from the deck. Find the probability of these events:
a) Each student draws a club.
b) Karen draws a red card, Gavin draws a king, Nasra draws a black card, Ali draws the 2 of clubs.
C) Karen draws a heart, Gavin draws a heart, Nasra draws a face card, and Ali draws an ace.
I don't understand this please help!
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events. How many of each are in the deck?
a) (13/52)^4 = ?
b) (26/52)(4/52)(26/52)(1/52) = ?
c) I'll let you do this one.
Sure, I can help you with that! Let's break down each part of the question and calculate the probability step by step.
a) Each student draws a club.
To calculate the probability of each student drawing a club, we need to know the total number of clubs in the deck and the total number of cards in the deck.
In a standard deck of playing cards, there are 52 cards and 13 clubs. Each student randomly draws one card from the deck without replacement, meaning once a card is drawn, it is not put back into the deck.
To calculate the probability, we'll start with the first student. The probability of the first student drawing a club is 13/52, or 1/4 since there are 13 clubs out of 52 cards in the deck.
For the second student, since one club has already been removed, there are 12 clubs out of 51 cards remaining in the deck. Therefore, the probability of the second student drawing a club is 12/51.
Repeat this process for the remaining two students. So the probability for each student is:
- Student 1: 13/52 (1/4)
- Student 2: 12/51
- Student 3: 11/50
- Student 4: 10/49
To calculate the probability that all four students draw a club, you multiply their individual probabilities together:
Probability = (13/52) * (12/51) * (11/50) * (10/49) = 0.006
Therefore, the probability that each student draws a club is 0.006, or 0.6%.
b) Karen draws a red card, Gavin draws a king, Nasra draws a black card, Ali draws the 2 of clubs.
To calculate this probability, we need to consider the number of possible outcomes that match the given conditions and divide it by the total number of outcomes.
- Karen drawing a red card: There are 26 red cards in a deck of 52, so the probability is 26/52 or 1/2.
- Gavin drawing a king: There are 4 kings in a deck of 52, so the probability is 4/52 or 1/13.
- Nasra drawing a black card: There are 26 black cards in a deck of 52, so the probability is 26/52 or 1/2.
- Ali drawing the 2 of clubs: There is only 1 2 of clubs in a deck of 52, so the probability is 1/52.
To calculate the overall probability, we multiply the individual probabilities together:
Probability = (1/2) * (1/13) * (1/2) * (1/52) = 0.000096
Therefore, the probability that Karen draws a red card, Gavin draws a king, Nasra draws a black card, and Ali draws the 2 of clubs is 0.000096 or 0.0096%.
c) Karen draws a heart, Gavin draws a heart, Nasra draws a face card, and Ali draws an ace.
In this scenario, we need to calculate the probability of each event separately and then multiply them together.
- Karen drawing a heart: There are 13 hearts in a deck of 52, so the probability is 13/52 or 1/4.
- Gavin drawing a heart: After Karen draws a card, there are only 12 hearts left out of 51 cards, so the probability is 12/51.
- Nasra drawing a face card: There are 3 face cards (King, Queen, and Jack) in each suit, totaling 12 face cards in a deck of 52. So the probability is 12/52 or 3/13.
- Ali drawing an ace: There are 4 aces in a deck of 52, so the probability is 4/52 or 1/13.
To find the overall probability, we multiply the individual probabilities together:
Probability = (1/4) * (12/51) * (3/13) * (1/13) = 0.001042
Therefore, the probability that Karen draws a heart, Gavin draws a heart, Nasra draws a face card, and Ali draws an ace is 0.001042 or 0.1042%.