two cards are drawn at random (without replacement) from a regular deck of 52 cards
a) what is the probability that the first card is a heart and the second is red?
b) what is the probability that the first card is red and the second is a heart?
the probability is that there are 2 out of 52 hearts
the other probability is that there are also 2 out 52 red hearts that you will get
a)
you will have to do this in 2 cases:
1. sequence is red heart, then red
prob = (1/52)(12/51)
2. sequence is non-red heart, then red
prob = (3/52)(13/51)
so prob = (1/52)(12/51) + 3/52)(13/51)
= 12/2652 + 39/2652
= 51/2652
= 1/52
b) again, use the same argument,
it could be ..
-the red-heart, then one of the other 3 hearts, or
- one of the other reds, then one of the hearts.
prob = (1/52)(3/51) + 12/52)(4/51)
= 1/52
To calculate the probabilities of drawing two cards with specific attributes, we need to understand the concept of probability and basic counting principles. Let's solve each part separately:
a) Probability that the first card is a heart and the second is red:
Step 1: Determine the total number of cards in the deck. A standard deck has 52 cards.
Step 2: Calculate the probability of drawing a heart as the first card. There are 13 hearts in a deck, so the probability of drawing one is 13/52 (since there are 52 cards in total).
Step 3: Once the first card is drawn, there are now 51 cards remaining in the deck.
Step 4: Calculate the probability of drawing a red card as the second card. There are 26 red cards in a deck (13 hearts + 13 diamonds), so the probability is 26/51 (as there are now 51 cards left in the deck).
Step 5: Multiply the probabilities from steps 2 and 4 to find the probability of drawing a heart first and then a red card:
(13/52) * (26/51) = 338/2652 ≈ 0.128 = 12.8%
Therefore, the probability that the first card drawn is a heart and the second card is red is approximately 12.8%.
b) Probability that the first card is red and the second is a heart:
Step 1: Determine the total number of cards in the deck. A standard deck has 52 cards.
Step 2: Calculate the probability of drawing a red card as the first card. There are 26 red cards in a deck, so the probability is 26/52 (since there are 52 cards in total).
Step 3: Once the first card is drawn, there are now 51 cards remaining in the deck.
Step 4: Calculate the probability of drawing a heart as the second card. There are still 13 hearts in a deck, so the probability is 13/51 (as there are now 51 cards left in the deck).
Step 5: Multiply the probabilities from steps 2 and 4 to find the probability of drawing a red card first and then a heart:
(26/52) * (13/51) = 338/2652 ≈ 0.128 = 12.8%
Therefore, the probability that the first card drawn is red and the second card is a heart is also approximately 12.8%.