Two cards are drawn from a standard deck of cards, and one of the two cards is noted and removed. Find the following probabilities of the second card.
P(spade | spade) =
P(red | red) =
The probability of both/all events occurring is determined by multiplying the probabilities of the individual events. If the first card is not replaced,
13/52 * 12/51 = ?
26/52 * 25/51 = ?
Mike,
Can you double check if your notation P(spade|spade) means
P(spade followed by a spade), usually written as P(spade,spade)
or does it means conditional probability
P(spade given it is a spade)?
To find the probabilities P(spade | spade) and P(red | red), we need to understand conditional probability.
Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, we want to find the probability of drawing a specific card (such as a spade or a red card) given that the first card drawn is a specific card (also a spade or a red card).
Let's break down the problem step by step:
Step 1: Determine the sample space
The sample space is the total number of possible outcomes.
In this case, we have a standard deck of 52 cards. So, the sample space is 52 cards.
Step 2: Determine the number of favorable outcomes
The favorable outcomes are the number of desired cards that we want to draw.
Now let's calculate each probability:
P(spade | spade):
The notation "P(spade | spade)" means the probability of drawing a spade given that the first card drawn is a spade.
In this scenario, the first card is a spade and has been removed from the deck. We are now drawing the second card.
Step 1: In the sample space, there are now 51 cards remaining.
Step 2: The number of favorable outcomes is 12. There are 12 spades left in the deck.
Therefore, P(spade | spade) = number of favorable outcomes / sample space = 12/51.
P(red | red):
The notation "P(red | red)" means the probability of drawing a red card given that the first card drawn is a red card.
In this scenario, the first card is a red card and has been removed from the deck. We are now drawing the second card.
Step 1: In the sample space, there are now 51 cards remaining.
Step 2: The number of favorable outcomes is 26. There are 26 red cards left in the deck (13 hearts + 13 diamonds).
Therefore, P(red | red) = number of favorable outcomes / sample space = 26/51.
To summarize:
- P(spade | spade) = 12/51
- P(red | red) = 26/51