two cards are drawn in succession without replacement from a standard deck of 52 cards. what is the probability that the first card is a spade given that the second card is a club. what is the probability that the first card is a face card(jack, queen, king) given that the second card is an eight? Please help, I've gotten .064 for the first one and .018 for the second one and both are wrong. I don't know how they could be wrong.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
Spade = 13/52, club = 13/(52-1)
face = 12/52, 8 = 4/(52-1)
I don't know how they could be wrong either. Text typos?
To find the probabilities in these scenarios, we need to apply conditional probability concepts.
1) Probability that the first card is a spade, given that the second card is a club:
We know that the second card is a club, so the sample space for the second card is reduced to 51 cards. Out of these 51 cards, there are still 13 spades remaining. Therefore, the probability that the first card is a spade, given that the second card is a club, can be calculated as:
P(First card is a spade | Second card is a club) = [P(First card is a spade and Second card is a club)] / P(Second card is a club)
The probability that the first card is a spade and the second card is a club can be calculated as follows:
P(First card is a spade and Second card is a club) = (13/52) * (13/51)
The probability that the second card is a club can be calculated as:
P(Second card is a club) = (13/52) + (13/51) + (13/50) + ... + (13/3) + (13/2) + (13/1) = 13/52
Putting it all together:
P(First card is a spade | Second card is a club) = [(13/52) * (13/51)] / (13/52) = 13/51
Therefore, the probability that the first card is a spade, given that the second card is a club, is 13/51.
2) Probability that the first card is a face card (jack, queen, king), given that the second card is an eight:
Similarly, we need to use conditional probability to calculate this.
P(First card is a face card | Second card is an eight) = [P(First card is a face card and Second card is an eight)] / P(Second card is an eight)
The probability that the first card is a face card and the second card is an eight can be calculated as follows:
P(First card is a face card and Second card is an eight) = (12/52) * (4/51)
The probability that the second card is an eight can be calculated as:
P(Second card is an eight) = (4/52) + (4/51) + (4/50) + ... + (4/3) + (4/2) + (4/1) = 4/52
Putting it all together:
P(First card is a face card | Second card is an eight) = [(12/52) * (4/51)] / (4/52) = 1/17
Therefore, the probability that the first card is a face card (jack, queen, king), given that the second card is an eight, is 1/17.
So, the correct answers are:
1) Probability that the first card is a spade, given that the second card is a club: 13/51.
2) Probability that the first card is a face card (jack, queen, king), given that the second card is an eight: 1/17.
I hope this explanation clarifies your doubts and helps you understand the correct approach to solving these probability problems.