I can not seem to figure out this question. Can someone please help?
Two cards are drawn from an ordinary deck of 52 playing cards with replacement. What is the probability that A) both cards are the same color? B) both cards are from the same suit? C) How would your answers to parta (a) and (b) change if the draws are made without replacement?
Thanks for your help.
A) half of the cards are red and half are blue. The probability is 1/4 that both are read and 1/4 that both are black. Add those and you get 1/2
B) Follow the same logic. Both cubs = 1/4. Both hearts = 1/4 etc. Add them up
C) What do tou think?
So would the answer to B)= 1/2.
1/2 for (B) is not correct.
"B) Follow the same logic. Both cubs = 1/4. Both hearts = 1/4 etc. Add them up "
Means you'd have to do some calculations for each of the 4 suits.
First club = 13/52
Second club = 13/52 (with replacement)
So both clubs would be 13/52*13/52=1/16
Same goes for each of the other three suits.
Add up the numbers for all four suits and you'll get the required answer.
C) Without replacement, the chance of getting the same color on two consecutive draws equals the probability of getting red twice in a row PLUS the probability of getting black twice in a row. That probability is 2*(1/2)*(25/51) = 25/51
The probability of getting two of the same suit in consecutive draws is
4*(1/4)*(12/51)= 12/51
The number 4 in front is due to the fact that there are four different suits; it could be any of them.
3. From a deck of 52 ordinary playing cards, two cards are drawn with replacement. Find the probability that both are hearts.
it would be 27/625
Sure, I can help you with this. Let's break down each part of the question:
A) What is the probability that both cards are the same color?
To solve this, first, we need to determine the number of ways we can choose two cards from a deck of 52 cards with replacement. Since we are choosing with replacement, it means after each draw, the card is placed back in the deck before the next draw. This means we still have 52 cards to choose from for the second draw.
The number of ways to choose two cards from a deck of 52 with replacement is calculated as 52 * 52 = 2,704.
Now, let's calculate the number of favorable outcomes. We have two possibilities for this:
1. Both cards are red: There are 26 red cards in a deck, so the probability of drawing a red card on the first draw is 26/52, and the probability of drawing another red card on the second draw is also 26/52. The probability of both cards being red is (26/52) * (26/52) = 1/4.
2. Both cards are black: Similarly, the probability of drawing a black card on the first draw is 26/52, and the probability of drawing another black card on the second draw is also 26/52. The probability of both cards being black is (26/52) * (26/52) = 1/4.
Now, add the probabilities of both favorable outcomes: 1/4 + 1/4 = 1/2.
Therefore, the probability that both cards are the same color is 1/2.
B) What is the probability that both cards are from the same suit?
To solve this, we need to determine the number of ways we can choose two cards of the same suit from a deck of 52 cards with replacement.
Since there are 4 suits in a deck, the number of favorable outcomes is 4, as we can choose any of the 4 suits.
Similar to part A, the total number of ways to choose two cards from a deck of 52 with replacement is 52 * 52 = 2,704.
Therefore, the probability of drawing two cards from the same suit is 4/2,704, which simplifies to 1/676.
C) How do the answers change if the draws are made without replacement?
If the draws are made without replacement, the probability of each event will change because after the first card is drawn, there will be one less card in the deck for the second draw. Essentially, the probability will depend on the outcome of the first draw.
For part A, the probability that both cards are the same color without replacement will be affected by the first draw. After the first card is drawn, there will be one less card of that color in the deck, affecting the probability of drawing the same color on the second draw. To calculate this probability, you would multiply the probability of drawing a card of a specific color on the first draw by the probability of drawing the second card of the same color from the reduced deck.
Similarly, for part B, the probability that both cards are from the same suit without replacement will be affected by the first draw. After the first card is drawn, there will be one less card of that suit in the deck, affecting the probability of drawing the second card of the same suit. Again, to calculate this probability, you would multiply the probability of drawing a card of a specific suit on the first draw by the probability of drawing the second card of the same suit from the reduced deck.
Keep in mind that when doing calculations without replacement, the probabilities change with each draw.